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A  COURSE  . 
IN  .  .  . 

ELEMENTARY 
MECHANICAL 
DRAWING  .  .. 


REVISED 
BY 

WM.  A.  PIKE, 
UNIVERSITY  OF 
MINNESOTA, 
MINNEAPOLIS, 
1891. 


TRIBUNE     JOB     PTG.    CO. 


Start 

Hfinex 

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PREFACE  TO  REVISED  EDITION. 


THIS  edition  has  been  revised  and  corrected  to  meet  objections 
found  in  the  first.  The  number  of  plates  of  geometrical  and 
projection  problems  have  been  reduced  and  more  problems  put 
on  each  plate,  as  experience  showed  that  too  much  time  was  taken 
'in  the  lettering  and  other  routine  work  of  so  many  plates;  also, 
as  some  of  the  problems  are  but  modifications  of  others  and  some 
seemed  of  hardly  enough  value  to  be  retained,  a  number  of  them 
have  been  dropped  in  order  to  give  more  time  to  the  application 
to  practical  drawings.  The  plates  have  been  redrawn  and  printed 
with  black  lines  on  a  white  ground  for  the  sake  of  greater  plain- 
ness. The  example  of  a  rough  sketch  from  which  to  make  a  fin- 
ished drawing  has  been  corrected  and  a  more  modern  stub  end 
substituted  for  the  old  one.  Additional  plates  illustrating  line 
shading  have  been  incorporated  in  the  text,  as  most  work  in 
actual  practice  is  finished  in  this  way  instead  of  by  tinting  and 
shading  with  the  brush.  Finally  the  text  has  been  changed  and 
added  to  wherever  it  seemed  desirable. 
MINNEAPOLIS,  Aug.  30th,  '91. 


2068014 


COURSE  IN 

Elementary   JVleehanieal  Drawing. 


DRAWING  MATERIALS  AND  INSTRUMENTS. 

Each  Student  will  require  the  following  Instruments  on  begin- 
ning the  course,  viz : — 

Half-a-dozen  Sheets  of  Drawing  Paper,  a  Drawing-board,  a 
T-square,  a  Pair  of  Triangles,  a  Hard  Pencil,  a  Right  Line  Pen, 
a  Pair  of  Compasses  with  Pen,  Pencil  and  Needle  Points,  a  Pair 
of  Plain  Dividers,  an  accurate  and  finely  divided  Scale,  a  piece 
of  India  Ink,  a  Rubber,  an  Irregular  Curve,  and  half-a-dozen 
Thumb-tacks. 

These  instruments  and  materials  are  all  that  are  absolutely 
required  up  to  the  time  of  commencing  tinting  and  shading,  when 
a  few  other  articles  will  be  needed,  which  will  be  spoken  of  in  their 
proper  places. 

Before  purchasing,  the  following  directions  about  the  different 
instruments  and  materials  should  be  noticed. 

PAPER. — For  all  the  drawings  of  this  course,  use  Whatman's 
Imperial  drawing  paper.  It  comes  in  sheets  of  convenient  size, 
and  is  well  adapted  to  the  work  of  the  course.  Six  sheets  will  be 
enough  up  to  the  time  of  tinting. 

DRAWING-BOARD. — Great  care  should  be  taken  to  secure  a  good 
drawing-board.  The  best  boards  are  those  made  of  thoroughly 
seasoned  white  pine,  one  inch  thick,  with  cleats  at  the  ends  flush 
with  the  surface  of  the  board.  The  most  convenient  size  is 
twenty-three  by  thirty-one  inches.  This  gives  a  small  margin  out- 
side of  a  whole  sheet  of  Imperial  paper,  which  is  twenty-two  by 
thirtv  inches. 


One  surface  of  the  drawing-board  must  be  plane,  and  the  edge 
from  which  the  T-square  is  used  must  be  perfectly  straight. 

T-SQI-ARE.— All  horizontal  lines  in  the  drawings  are  made  by 
the  use  of  the  T-square.  The  T-square  should  be  used  from  the 
left-hand  edge  of  the  board,  unless  the  person  is  left-handed,  in 
which  case  it  should  be  used  from  the  right  hand  edge.  The  upper 
edge  of  the  blade  only  is  to  be  used  in  drawing  lines.  The  blade 
should  be  at  least  thirty  inches  long,  and  about  two-and-one-half 
inches  wide.  The  thickness  should  not  be  over  an  eighth  of  an 
inch.  The  head  should  be  twelve  or  fourteen  inches  long,  at  least, 
in  order  that  the  blade  may  never  be  thrown  out  of  line.  By  slid- 
ing the  head  up  and  down  on  the  straight  edge  of  the  board,  any 
number  of  parallel  horizontal  lines  may  be  drawn.  It  is  very 
important  that  the  upper  edge  of  the  T-square  be  perfectly 
straight. 

TRIANGLES. — For  making  all  vertical"  lines,  and  all  lines  mak- 
ing the  angles  of  thirty,  forty-five  and  sixty  degrees  with  the  hori- 
zontal and  vertical  lines,  triangles  are  used,  sliding  on  the  upper 
edge  of  the  T-square.  Two  triangles  are  necessary,  one  forty-five 
degree  and  one  thirty  and  sixty  degree,  as  they  are  called  from 
their  angles.  Bach  of  these  triangles  has  one  right  angle,  and 
either  can  l>e  used  for  drawing  verticals.  It  is  often  convenient  to 
have  one  triangle  large  enough  for  drawing  quite  long  verticals, 
like  border  lines ;  but  in  lettering  and  in  other  small  work  a  small- 
er one  is  much  more  convenient.  It  is  therefore  advisable  to  get  a 
thirty  and  sixty  degree  triangle  that  has  one  of  its  rectangular 
edges  about  ten  inches  long,  and  to  get  a  forty-five  degree  triangle 
much  smaller. 

PENCILS. — Ail  lines  are  to  be  made  first  with  a  hard  pencil,  and 
afterwards  to  be  inked.  It  is  very  important  that  the  pencil  lines 
be  very  fine  and  even,  though  they  need  not  be  very  dark. 

Ink  will  not  run  well  over  a  soft  pencil  line,  and  it  is  impossible 
to  do  good  work  without  making  the  lines  fine.  The  best  pencils 
for  this  work  are  Faber's  H  H  H  H  and  H  H  H  H  H  H  or  some 
kind  equally  hard  and  even.  The  H  H  H  H  is  recommended  for 
beginners  who  are  not  accustomed  to  using  a  very  hard  pencil, 
but  the  H  H  H  H  H  H  is  harder,  and  better  adapted  for  nice  work. 
One  of  each  kind  will  be  amply  sufficient  for  the  work  of  the 
whole  course.  The  pencil  should  be  sharpened  at  both  ends,  at 
one  end  with  a  com  in  on  sharp  round  point,  and  at  the  other  with 
the  lead  of  about  the  shape  of  the  end  of  a  table  knife.  The  round 


*  By  vertical  lines  are  meant  lines  parallel  to  the  edge  of  the  board 
from  which  the  T-square  is  tised,  by  horizontal  those  parallel  to  the  up- 
per edge  of  the  T-square. 


point  is  to  be  used  in  lettering  and  in  other  small  work,  and  the 
flat  point  in  making  long  lines.  The  flat  point  will  keep  sharp 
much  longer  than  a  round  point.  Both  points  should  be  sharpen- 
ed often  by  rubbing  them  on  a  piece  of  fine  sand  paper  or  on  a  very 
fine  file.  The  flat  point  should  always  be  used  in  the  compasses 
with  the  edge  perpendicular  to  the  radius  of  the  circle. 

RIGHT-LINE  PEN. — In  selecting  a  right-line  pen  care  should  be 
taken  to  get  one  with  stiff  nibs,  curved  but  little  above  the  points. 
If  the  nibs  are  too  slender  they  may  bend  when  used  against  the 
T-square  or  triangles,  and  the  result  will  be  an  uneven  line.  If  the 
nibs  are  too  open  there  is  danger  of  the  ink  dropping  out  and  mak- 
ing a  blot.  If  too  little  curved  the  pen  will  not  hold  ink  enough. 
The  nibs  are  apt  to  be  too  open,  than  otherwise.  The  medium 
sized  pens  are  best  adapted  for  this  work.  The  pen  must  have  a 
good  adjustment  screw  to  regulate  the  width  of  the  lines.  The 
pens,  as  they  are  bought,  are  generally  sharpened  ready  for 
use;  but,  after  being  used  for  a  time,  the  ends  of  the  nibs  get  worn 
down,  so  that  it  is  impossible  to  make  a  smooth,  fine  line.  When 
this  occurs  they  should  be  sharpened  very  carefully  on  a  fine  stone. 
In  order  to  have  a  pen  run  wrell,  two  things  are  necessary,  first  the 
points  must  be  exactly  the  same  shape  aud  length,  and  both  nibs 
must  be  sharp.  In  sharpening  a  pen,  therefore,  the  first  thing  to 
be  done  is  to  even  the  points.  This  may  be  done  1nr  moving  the 
pen  with  a  rocking  motion  from  right  to  left  in  a  plane  perpen- 
dicular to  the  surface  of  the  stone  while  the  nibs  are  screwed 
together.  After  the  nibs  are  evened  in  this  way  the  points  should 
be  opened  and  each  nib  sharpened  on  the  outside,  only,  by  holding 
the  pen  at  an  angle  of  about  thirty  degrees  with  the  surface  of  the 
stone,  while  it  is  moved  in  about  the  same  manner  as  in  sharpen- 
ing a  gouge.  The  point  should  be  examined  often  with  a  lens. 

COMPASSES. — The  compasses  must  have  needle  points,  with 
shoulders  to  prevent  them  from  going  into  the  paper  below:  a 
certain  depth.  The  needle  point,  when  properly  used,  leaves  a 
very  slight  hole  in  the  center  of  each  circle;  while  the  triangular 
point  with  which  the  poorer  instruments  are  provided,  leaves  a 
very  large,  unsightly  hole,  unless  used  with  more  than  ordinary 
care.  The  pencil  point  should  be  one  made  to  contain  a  small 
piece  of  lead  only.  All  that  has  been  said  in  regard  to  the  right- 
line  pen  applies  equally  well  to  the  pen  point  of  the  compass.  In 
using  the  pen  point  be  sure  that  both  nibs  press  equally  on  the 
paper,  otherwise  it  will  be  impossible  to  make  an  even  line.  Both 
nibs  may  be  made  to  bear  equally  by  adjusting  the  points  in  the 
legs  of  compasses. 

DIVIDERS. — The  dividers  should  be  separate  from  the  com- 
passes, as  it  is  very  inconvenient  to  be  obliged  to  change  the 


points  whenever  the  dividers  are  needed.  The  dividers  have  tri- 
. insular  points,  which  should  be  very  fine,  and  of  the  same  length. 
The  legs  of  the  dividers  should  move  smoothly  in  the  joint,  and 
not  hard  enough  to  cause  them  to  spring  while  being  moved.  The 
dividers  are  used  principally  for  spacing  oft"  equal  distances  on 
lines,  but  are  often  used  for  taking  measurements  from  the  scale, 
especially  when  the  same  measurement  is  to  be  used  on  several 
different  parts  of  a  drawing. 

SCALE. — A  ver3'  good  scale,  for  this  course,  is  one  with  inches 
divided  into  fourths, eighths,  sixteenths. etc.,  on  one  edge;  and  into 
twelfths,  twenty -fourths,  etc.,  on  the  other.  The  first  edge  is  verv 
convenient  for  taking  measurements,  and  for  making  drawings  to 
a  scale  of  one-half,  one-fourth,  etc.;  but  the  second  is  better  for 
drawing  to  a  scale  of  a  certain  number  of  inches  to  the  foot. 
Triangular  scales  are  still  better,  but  more  expensive. 

INK.— India  ink,  which  comes  in  sticks,  is  the  best  ink  tor  gen- 
eral uses ;  but  there  are  several  kinds  of  ink  in  bottles  which  are 
much  more  convenient  for  making  line  drawings.  None  of  the  ink 
that  comes  in  bottles,  however,  is  good  for  shading.  If  the  India' 
ink  is  used,  an  ink  slab  or  saucer  will  be  needed  in  addition  to  the 
instruments  mentioned  in  the  list.  In  grinding  India  ink,  a  small 
quantity  of  water  is  sufficient,  and  the  ink  should  be  ground  until 
a  very  fine  line  can  be  made  very  black  with  one  stroke  of  the  pen. 
Ink  will  look  black  in  the  slab  long  before  it  is  fit  to  use  on  adraw- 
ing.  Ink  should  not  be  ground,  however,  so  thick  that  it  will  not 
run  well  in  the  pen.  The  ink  must  be  kept  covered  up  or  it  will 
soon  evaporate  so  much  as  to  be  too  thick  to  run  well. 

Mi  HBER. — Get  a  soft  piece  of  rubber  so  as  not  to  injure  the 
surface  of  the  paper  in  rubbing;  what  is  known  as  velvet  rubber 
is  well  adapted  to  the  draughtsman's  use. 

IRREGTLAR  CURVE.— In  selecting  an  irregular  curve,  one  should 
be  obtained  which  has  very  different  curvature  in  different  parts, 
in  order  to  fit  curves  which  cannot  be  drawn  with  compasses. 

THUMB  TACKS.— Thumb  tacks  should  have  good  large  heads, 
so  firmly  fastened  on  that  they  cannot  get  loose. 

One  can  do  much  better  in  buying  instruments,  to  buy  them 
in  separate  pieces,  each  carefully  selected,  than  to  buy  them  in 
sets.  It  is  very  difficult  to  buy  a  set  of  instruments  that  will  con- 
tain just  what  is  required  for  this  work,  without  buying  many 
unnecessary  pieces. 


'03 


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GENERAL  DIRECTIONS  FOR  COMMENCING  THE 
WORK. 

Each  plate  of  geometrical  problems  is  to  be  made  on  a  half 
sheet  of  the  Imperial  paper.  The  sheet  should  be  folded  over  and 
cut  with  a  sharp  knife,  but  before  cutting  find  out  which  is  the 
right  side  of  the  paper.  The  right  side  of  Whatman's  paper  may 
always  be  found  by  holding  the  sheet  up  to  the  light.  When  the 
name  of  the  manufacturer  can  be  read  from  left  to  right,  the  right 
side  is  the  one  toward  the  holder.  The  half  that  has  not  the 
name  on  it  should  be  used  first,  while  its  right  side  is  known ;  the 
right  side  of  the  other  piece  can  be  found  in  the  way  described, 
when  it  is  to  be  used. 

Place  the  paper  on  the  drawing  board  so  that  two  of  its  edges 
will  be  parallel  to  the  upper  edge  of  the  T-square  when  in  position 
on  the  edge  of  the  board  ;  and  fasten  it  down  with  three  thumb- 
tacks in  each  of  the  long  sides,  placing  each  thumb  tack  within  a 
cjuarter  of  an  inch  of  the  edge,  in  order  that  the  holes  may  be  cut 
off  when  the  plate  is  trimmed.  For  convenience  in  working  on  the 
upper  part  of  the  plate,  it  is  best  to  have  the  paper  as  near  the 
bottom  of  the  board  as  possible. 

Begin  each  plate  by  drawing  a  horizontal  line,  with  the  use  of 
the  T-square,  as  near  the  thumb  tacks  at  the  top  as  possible. 
Fourteen  inches  below  the  first  line,  if  the  longest  dimension  is  to 
be  horizontal,  draw  another  parallel  to  it,  at  the  bottom  of  the 
paper,  and  by  means  of  the  larger  triangle,  draw  vertical  lines  at 
the  right  and  left  of  the  paper,  twenty-one  inches  apart.  The  lines 
are  the  limits  of  the  plate,  and  are  the  ones  that  the  plate  is  to  be 
trimmed  by,  when  finished. 

All  the  plates  that  are  to  be  drawn  on  a  half-sheet  must  be  of 
this  size,  twenty-one  by  fourteen  inches,  unless  the  paper  is  to  be 
shrunk  down,  in  which  case  the  plates  must  be  made  somewhat 
smaller,  as  will  be  afterwards  noticed. 

All  the  plates  are  to  have  a  border  line  one  inch  from  the  fin- 
ished edge,  except  on  the  top,  where  the  border  is  to  be  one  and  a 
quarter  inches  from  the  edge.  This  border  should  next  be  drawn 
by  spacing  off  the  proper  distances  from  the  lines  just  drawn,  and 
drawing  the  border  with  T-square  and  triangles. 

There  are  to  be  fifteen  geometrical  problems  to  each  plate,  and 
for  convenience  in  locating  them,  the  space  inside  of  the  border 
lines,  in  the  first  five  plates,  should  be  divided  into  fifteen  equal 


10 

rectangles,  in  three  rows  of  five  each.    These  last  lines  are  not  to 
be  inked,  but  must  1>e  erased  when  the  plate  is  completed. 

The  first  two  plates  in  the  course  are  of  geometrical  problems. 
The  problems  that  have  been  selected  have  many  applications  in 
subsequent  work  and,  moreover,  the  exact  construction  of  them 
-ivc<  the  best  of  practice  for  beginners  in  handling  the  different 
instruments.  The  construction  of  each  problem  is  described  in  the 
text,  with  references  to  the  plates;  and  each  must  be  constructed 
according  to  the  directions.  The  reasons  for  the  different  con- 
structions, though  necessarily  omitted  in  the  text,  will  l>e  evident 
to  every  one  who  has  a  knowledge  of  plane  geometry. 

The  geometrical  problems  are  not  to  be  drawn  to  scale,  but 
they  should  be  so  proportioned  that  they  will  occupy  about  the 
same  amount  of  space  in  the  center  of  each  rectangle. 

All  of  the  lines  must  be  made  very  fine  and  even ;  and  great 
care  must  be  taken  to  get  good  intersections  and  tangencies. 

DIRECTIONS   FOR   LETTERING. 

Alter  the  problems  are  pencilled  they  must  be  lettered  to 
correspond  to  the  plates  in  this  pamphlet.  Make  all  the  letters 
on  the  plates  of  geometrical  problems  and  elementary  projections 
like  those  given  in  Plate  A.  These  skeleton  letters  are  the  sim- 
plest of  all  mechanical  letters  to  construct,  and,  when  well  made, 
they  are  more  appropriate  for  such  work  than  if  more  elaborate. 
Make  the  small  letters,  in  every  case,  two-thirds  as  high  as  capi- 
tals. 

Before  making  a  letter  draw  a  small  rectangle  that  will  just 
contain  the  letter,  and  then  construct  the  letter  within  the  rec- 
tangle, as  shown  in  plate  A,  using  instruments  wherever  possible. 
The  height  of  all  the  capital  letters  in  the  problems  and  in  the 
general  title  at  the  top,  is  to  be  one  quarter  of  an  inch.  The 
widths  vary,  and  may  l)est  be  found  in  each  case,  until  practice 
renders  it  unnecessary,  by  consulting  plate  A.  Great  care  must 
be  taken  in  lettering  to  make  all  the  lines  of  the  letters  of  the 
same  si/e,  and  in  joining  the  curves  and  straight  lines. 

TITLES. 

The  title  of  each  plate  of  geometrical  problems  must  corre- 
spond to  that  given  in  plate  1,  except  as  to  number.  The  title  of 
the  projection  plates  will  correspond  to  that  of  plate  III, 
and  the  titles  of  all  other  drawings  will  be  as  indicated  in 
the  text.  In  constructing  a  title  always  work  both  ways 
from  the  central  letter  of  the  title,  in  order  that  the  title  may  be 
symmetrical,  and  over  the  center  of  the  plate.  In  order  to  find 


11 

the  middle  letter  of  the  title,  count  the  number  of  letters,  consider- 
ing the  space  between  words  as  equal  to  that  of  a  letter,  and  di- 
vide the  number  of  spaces  thus  found  by  two ;  this  will  give  the 
number  of  the  middle  letter  from  either  end  of  the  title.  Con- 
struct this  letter  .over  the  centre  of  the  plate,  and  then  work  both 
ways  from  this  in  the  wa3r  just  indicated.  Make  the  letters  in  a 
word  about  an  eighth  of  an  inch  apart,  though  the  space  will 
vary  with  the  shape  of  the  letter ;  and  the  space  between  words 
equal  to  that  of  an  average  letter  with  its  spaces. 

It  is  best,  in  all  cases,  to  have  the  title  before  you  in  rough  let- 
ters, to  avoid  making  mistakes  in  working  backwards  from  the 
middle  letter.  The  titles  at  the  top  are  to  be  made  in  capitals. 
The  letters  in  the  general  title  are  to  be  a  quarter  of  an  inch  high 
and  a  quarter  of  an  inch  above  the  border,  and  those  in  number 
of  the  plate  of  letters  three-sixteenths  of  an  inch  high  and  the 
same  distance  above  the  general  title. 

The  name  of  the  draughtsman  should  be  in  the  first  three 
plates,  at  the  lower  left  hand  corner,  three-sixteenths  of  an  inch 
below  the  border  and  the  date  of  completion  in  a  corresponding 
position  at  the  right.  Make  the  date  first,  and  commence  the 
name  as  far  from  the  edge,  at  the  left,  as  the  first  figure  of  the  date 
comes  from  the  right-hand  edge.  Make  the  capitals  in  name  and 
date  three-sixteenths  of  an  inch  high. 

Number  the  problems  as  they  are  in  the  plates,  commencing 
the  first  letter  of  the  abreviations  for  problems  in  capitals,  one- 
half  an  inch  below,  and  half  an  inch  to  the  right  of  the  lines  form- 
ing the  upper  right  hand  corner  of  the  rectangle.  The  other 
letters  of  the  abbreviations  are  to  be  small,  and  the  numbers  of 
the  problems  are  to  be  marked  with  figures  of  the  same  height  as 
the  capitals. 

Great  pains  must  be  taken  in  lettering  the  plates,  as  the  gen- 
eral appearance  of  a  drawingis  very  much  affected  by  the  arrange- 
ment and  construction  of  the  letters  and  titles.  The  directions 
here  given  apply  to  the  plates  of  geometrical  problems.  Some 
modifications  will  be  made  in  lettering  the  problems  in  projection ; 
but  the  remarks  on  construction  of  the  separate  letters,  and  on 
the  arrangement  of  the  letters  in  a  title,  are  general.  After  having 
had  the  practice  in  spacing  and  proportioning  the  skeleton  letters, 
in  the  first  three  plates,  the  student  will  be  allowed  to  use  other 
styles  of  letters  on  the  remaining  work.  Care  must  be  taken, 
however,  to  have  the  titles  symmetrical,  and  no  letters  on  the 
plates  of  this  course  should  be  made  over  half  an  inch  high. 

INKING. 

When  the  lettering  is  all  done,  a  plate  is  ready  to  be  inked.  Be- 
fore using  the  pen  on  the  plate,  be  sure  that  it  is  in  a  condition  to 


12 

make  a  fine,  even  line,  by  testing  it  on  a  piece  of  drawing  paper  or 
on  the  part  of  your  paper  that  is  to  be  trimmed  off.  Be  sure  to 
have  ink  enough  ground  to  ink  the  whole  plate,  as  it  is  not  best  to 
change  the  ink  while  working  on  a  plate,  for  the  reason  that  it  is 
nearly  impossible  to  get  the  second  lot  of  the  same  shade  and 
thickness  as  the  first.  The  arcs  of  circles  should  be  inked  first,  for 
it  is  easier  to  get  good  intersections  and  tangencies  by  so  doing, 
than  it  is  if  the  straight  lines  are  drawn  first.  Make  all  the  given 
lines  and  all  the  required  lines  in  full;  but  all  the  construction  lines 
in  fine  dots.  Make  all  the  lines  in  the  geometrical  problems  as 
fine  and  even  as  possible.  The  border  lines  should  be  made  a  little 
heavier  than  the  others.  All  the  fine  lines  should  be  made,  if  pos- 
sible, with  one  stroke  of  the  pen.  In  order  than  an  even  line  may 
be  made,  the  pen  must  be  held  so  that  both  nibs  will  bear  on  the 
paper  equally ;  and  in  order  to  do  this,  the  T-square  or  triangle 
must  be  held  a  little  way  from  the  line,  but  parallel  to  it.  The  pen 
should  be  inclined  slightly  in  the  direction  it  is  to  be  moved. 

In  using  the  compass  pen,  the  joints  of  the  compass  legs 
should  lie  so  adjusted  that  the  point  will  bear  equally  on  both  nibs. 

The  ink  should  l>e  placed  in  the  pens  by  means  of  a  quill  or  a 
thin  sliver  of  wood.  The  pen  should  never  be  dipped  into  the  ink, 

THE  PLATES. 

The  plates  in  this  pamphlet  are  given  to  show  the  arrangement 
and  construction  of  the  problems  but  should  not  be  followed  as  ex- 
amples too  closely,  as  mechanical  difficulties  make  it  necessary  to 
use  coarser  lines  in  proportion  to  the  size  of  the  plates  than  should 
appear  on  the  drawings. 

GEOMETRICAL  PROBLEMS. 

PROBLEM  1. — To  bisect  a  given  line,  A  B,  or  to  erect  a  perpen- 
dicular at  the  middle  point  of  A  B. 

From  A  and  B  as  centres,  with  a  radius  greater  than  one-half 
of  A  B,  described  two  arcs  intersecting  at  C,  and  two  arcs  inter- 
secting at  D.  Join  C  and  D  by  a  straight  line,  it  will  bisect  A  B, 
and  will  be  perpendicular  to  it. 

PROBLEM  2.— To  divide  a  given  line,  A  B,  into  any  number  of 
equal  parts,  five  for  instance. 

Draw  a  line,  A  C,  making  any  angle  with  A  B,  and  on  A  C  set 
off  any  five  equal  distances,  A  1,  1  2,  2  3,  3  4  and  4  C ;  join  C  and 
B,  and  through  1,  2,  3  and  4,  draw  lines  paralled  to  C  B,  these 
lines  will  divide  A  B  into  eqmil  parts. 

PROBLEM  3.— To  draw  a  perpendicular  to  a  line  B  C,  from  a 
point  A,  without  the  line. 

From  A  as  a  centre,  and  with  any  radius,  describe  an  arc,  cut- 


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13 

ting  B  C  in  D  and  E.    From  D  and  E  as  centres,  describe  two  arcs 
intersecting  in  F.    Join  F  with  A. 

PROBLEM  4. — To  draw  a  perpendicular  to  a  line  B  C  from  a 
point  A,  nearly  over  one  end,  C,  so  that  problem  3  cannot  be  used. 

From  any  point  B,  on  the  given  line  as  a  centre,  describe  an  arc 
passing  through  A.  From  some  other  point  D,  of  B  C,  describe 
another  arc  passing  through  A.  Join  A  with  the  other  point  of 
intersection  of  the  arcs. 

PROBLEM  5. — To  erect  a  perpendicular  to  a  line  B  C,  at  a  given 
point  A,  of  the  line. 

Set  off  from  A,  the  equal  distances  A  E  and  A  F,  on  either  side. 
From  E  and  F  as  centres,  with  any  radius  greater  than  A  E  and  A 
F;  describe  two  arcs,  intersecting  at  D.  Join  D  with  A. 

PROBLEM  6. — To  draw:  a  line  parallel  to  a  given  line  A  B,  at  a 
given  distance  from  A  B. 

From  two  points  C  and  D,  of  A  B,  which  should  not  be  too 
near  together,  describe  two  arcs,  with  the  given  distance  as  a 
radius.  Draw  a  line  E  F  tangent  to  these  arcs. 

PROBLEM  7. — To  lay  off  an  angle  at  a  given  point  a,  on  a  given 
line  a  c,  equal  to  a  given  angle  BAG. 

With  A  as  a  centre  and  any  radius,  describe  an  arc  included 
Ijetween  B  A  and  A  C.  With  a  as  a  centre  and  the  same  radius, 
describe  an  indefinite  arc.  Lay  off  the  chord  b  c  equal  B  C  from  c 
on  the  arc  b  c.  Join  b  with  a. 

PROBLEM  8. — To  bisect  a  given  angle  BAG,  whose  vertex  A  is 
within  the  limits  of  the  drawing. 

From  A  as  a  centre  describe  an  arc,  cutting  A  B  and  A  C  in  b 
and  a  respectively.  From  b  and  a  as  centres  describe  two  arcs  in- 
tersecting in  c.  Join  c  with  A. 

PROBLEM  9. — To  bisect  an  angle  BA — CD,  whose  vertex isnot 
within  the  limits  of  the  drawing. 

Draw  by  problem  7,  two  parallels,  ab  and  ac  to  AB  and  CD 
respective!}-,  and  at  the  same  distance  from  A  B  and  C  D ;  this  dis- 
tance must  IDC  such  that  ab  and  ac  shall  intersect.  The  problem 
is  then  reduced  to  one  of  bisecting  bac,  which  is  done  by  prob- 
lem 8. 

PROBLEM  10. — To  pass  the  circumference  of  a  circle  through 
three  points,  A,  B,  C. 

Draw  the  lines  A  B  and  B  C.  Bisect  A  B  and  B  C  by  perpen- 
diculars, by  problem  1.  With  D,  the  intersection  of  these  perpen- 

•The  reverse  of  this  problem  can  be  solved,  i.  e..  to  find  the  centre  of 
a  given  circle,  by  choosing  the  three  points  A,  B,  and  C  on  the  circumfer- 
ence of  the  circle. 


14 

cliculars.  as  a  mitre,  and  I)  A  as  a  radius,  describe  a  circumfer- 
i-iK-e.  it  will  pass  through  A,  H  and  C.* 

PKO»I.I.:M  1 1.— To  draw  two  tangents  to  a  circle,  whose  centre 
is  ( ),  from  a  point  A,  without  the  circle. 

loin  A  with  ();  on  O  A  as  diameter,  describe  a  circle.  Join  the 
points  B  and  C.  in  which  the  latter  circle  intersects  the  given  one, 
with  A.  A  B  and  A  C  will  be  the  required  tangents. 

PKOHI.I-: M  1  2.— To  draw  circles,  with  given  radii  c  and  d.  tan- 
gent internally  and  externally  resj>ectively  to  a  circle  whose  centre 
is  ( »,  at  a  given  point  A. 

First,  internally.  Join  A  with  O.  Lay  off  on  A  O  from  A,  A  a 
equal  to  «.-.  From  a  as  a  centre,  and  Aa  as  a  radius,  describe  a 
circle. 

Second,  externally.  Prolong  OA,  and  lay  off  from  A,  Ab 
equal  to  d.  With  b  as  a  centre  and  A  b  as  a  radius,  describe  a 
circle. 

PROBLEM  13.— To  draw  a  circle  with  a  given  radius  m,  tan- 
gent to  two  given  lines  AB  and  AC. 

Draw  a  1>  and  ac  parallel  to  A  B  and  A  C  respectively,  and  at 
a  distance  from  them  equal  to  m.  With  the  point  of  intersection 
a.  of  a  b  and  a  c  as  a  centre,  and  m  as  a  radius,  describe  a  circle. 

PKOHI.KM  14-.— To  draw  a  circle  with  a  given  radius  m,  tan- 
gcnt  to  a  given  circle  O,  and  to  a  given  line,  A  B. 

With  ( )  as  a  centre  and  O  a  equal  to  m  plus  the  radius  of  the 
given  circle,  as  a  radius,  describe  an  arc,  a  b.  Draw  the  line  C  c 
parallel  to  A  B,  and  at  a  distance  m  from  it,  by  problem  7.  With 
C,  the  intersection  of  the  arc  and  parallel,  as  a  centre,  and  m  as  a 
radius,  dcscrilie  a  circle. 

PROBLEM  15.— To  draw  a  circle  tangent  to  a  given  circle  O,  at 
a  given  point  A,  and  to  a  given  line  BC. 

Join  OA;  at  A  draw  the  tangent  AB,  perpendicular  to  OA, 
and  produce  it  till  it  meets  BC  at  B.  Bisect  the  angle  A  B  C,  by 
the  line  Ba,  by  problem  8.  Produce  B  a  till  it  meets  O  A  pro- 
duced in  D.  With  D  as  center,  and  DA.  a  radius,  describe  a  circle. 

PROBLEM  16.— To  draw  a  circle,  tangent  to  a  given  line  AB, 
at  a  given  point  C,  and  to  a  given  circle  O. 

At  C,  draw  DC,  perpendicular  to  A  B,  by  problem  5,  and  pro- 
duce DC  below  AB  till  Ca  is  equal  to  the  radius  of  the  given  cir- 
cle. Join  a  with  O,  and  by  problem  1,  erect  a  perpendicular  D  b  at 
the  middle  point  of  Oa.  With  the  intersection  D,  of  Da  and  Db, 
as  a  centre,  describe  a  circle. 

I'KOHI.KM  17.— To  draw  a  circle  tangent  to  a  given  circle  C, 
at  a  given  point  A,  and  to  a  given  circle  O. 


.J  "5 


15 

Join  AC,  and  produce  it  till  AD  is  equal  to  the  radius  of  the 
other  circle.  Join  D  with  O,  and  bisect  OD  by  a  perpendicular  Ea, 
by  problem  1.  With  E,  the  intersection  of  Ea  and  AD  produced 
as  a  centre,  and  E  A  as  a  radius,  describe  a  circle. 

PROBLEM  18. — Given  two  parallels,  B  A  and  CD,  to  draw  re- 
versed curve  which  shall  be  tangent  to  them  at  A  and  C. 

Join  AC.  Bisect  AC  in  2,  which  will  be  the  reversing  point. 
Bisect  B  2,  and  2  C  by  perpendiculars,  1  F  and  3  E.  Draw  AF 
and  C  E  perpendicular  to  B  A  and  C  D,  and  with  the  intersection 
E  of  C  E  and  3  E,  and  the  intersection  F  of  1  F  and  F  A,  as  cen- 
tres, and  radii,  equal  to  EC,  or  AF,  describe  two  arcs. 

PROBLEM  19. — Given  two  parallels  B  A  and  CD,  to  draw  a  re- 
versed curve,  whose  tangents  at  A  and  C  shall  be  perpendicular  to 
B  A  and  CD. 

Join  AC.  Bisect  AC  in  2,  wiich  will  be  the  reversing  point. 
Bisect  A  2,  and  2  C  by  perpendiculars  1 E  and  3F.  With  the  in- 
tersection D,  of  1  E  and  C  A.  and  the  intersection  F,  of  3  F  and  D 
C,  as  centres,  and  radii  equal  to  EA  or  FC,  describe  two  arcs. 

PROBLEM  20. — Given  a  circle  O,  to  inscribe  and  circumscribe 
regular  hexagons,  and  to  inscribe  a  regular  triangle. 

Lay  off  the  radius  O  A,  six  times  as  a  chord  on  the  circum- 
ference. For  the  circumscribed  hexagon,  draw  parallels  to  the 
sides  of  the  inscribed  figure,  which  shall  be  tangent  to  the  circle. 

Join  the  alternate  points  of  division  of  the  circle  for  the  in- 
scribed regular  triangle. 

PROBLEM  21.— Given  a  circle  O,  to  inscribe  and  circumscribe 
regular  octagons. 

Find  first  the  sides  A  B  and  C  of  an  inscribed  square,  by  con- 
necting the  ends  of  two  diameters  at  right  angles  to  each  other. 
Bisect  these  chords  by  perpendiculars,  and  thus  the  arcs  sub- 
tended by  them.  Join  the  points  C,  etc.,  with  the  vertices  of 
the  square  for  the  inscribed  octagon.  For  the  circumscribed  octa- 
gon, proceed  as  in  circumscribing  the  regular  hexagon  in  problem 
20. 

PROBLEM  22. — To  construct  a  regular  polygon  with  a  given 
number  of  sides,  five  for  instance,  the  sides  to  be  of  a  given  length 
AB. 

On  A  B  as  a  radius  describe  a  semi-circle.  Divide  the  semi-cir- 
cumference into  five  equal  parts,  A  4,  4  3,  and  so  on.  Omitting 
one  point  of  division  1,  draw  radii  through  the  remaining  points 
and  produce  them.  With  2  as  a  centre,  and  A  B  as  a  radius, 
describe  an  arc  cutting  B  3  produced  in  C;  B  2  and  2  C  will  be 
two  sides  of  the  polygon.  With  C  as  a  centre,  and  A  B  as  a  radius, 


16 

descrilx-  a.,  arc  cutting  B  4  produced  in  D;  C  D  will  be  another 
side.  Continue  this  construction;  the  last  point  should  come 
at  A. 

PROBLEM  23.— On  A  B  and  C  D  as  major  and  minor  axis,  to 
construct  an  ellipse. 

We  proceed  on  the  principle  that  the  sum  of  the  distances  of 
any  point  of  an  ellipse  from  the  foci  is  equal  to  the  major  axis. 
\\\  must  first  fix  the  position  of  the  foci.  From  C  as  a  centre,  and 
O  B  as  a  radius,  descrilx?  two  arcs,  cutting  A  B  in  a  and  b,  these 
are  the  foci.  To  apply  the  principle  just  mentioned,  take  the  dis- 
tance from  any  point,  as  c  of  A  B  to  A  and  B  as  radii,  and  a  and  b 
as  centres.  By  describing  arcs  above  and  below  A  B,  and  using 
both  radii  from  each  centre,  four  points  of  the  ellipse  will  be 
obtained.  Other  points  are  obtained  by  taking  other  points  on  A 
B,  and  proceeding  in  the  same  way.  Connect  the  points  found 
in  this  way  by  using  the  irregular  curve.  In  using  the  irregular 
curve  always  l>e  sure  to  have  it  pass  through  at  least  three  points. 

I'KOHLKM  24.— On  A  B  and  C  D  as  a  major  and  minor  axis,  to 
construct  an  ellipse;  another  method. 

On  the  straight  edge  of  a  slip  of  card  board  or  paper,  set  off 
three  points,  o,  c,  a,  the  distance  o  a  being  equal  to  the  given  semi- 
major  axis,  and  o  c  to  the  semi-minor.  Place  the  slip  in  various 
positions  such  that  a  shall  always  rest  on  the  minor,  and  c  on  the 
major  axis.  The  various  positions  marked  by  the  point  o  will  be 
points  of  the  ellipse. 

PKOIU.KM  U~>.-— To  construct  a  parabola,  given  the  focal  dis- 
tance O  E. 

We  proceed  on  the  principle  that  the  distance  of  any  point 
from  a  line  A  B,  called  the  directrix,  is  equal  to  its  distance  from  a 
certain  point  called  the  focus.  Draw  the  indefinite  line  A  B,  for  the 
directrix,  and  C  D  perpendicular  to  it.  From  C,  lay  off  C  E  and  E  O 
each  equal  to  the  focal  distance.  The  point  O  is  the  focus.  Draw 
a  number  of  perpendiculars  to  C  D  at  various  points.  To  find  the 
points  in  which  the  parabola  intersects  any  one  of  them  as  a  a', 
describe  an  arc  with  O  as  a  centre,  and  a  C,the  distance  from  that 
perpendicular  to  C  as  a  radius.  E  is  the  point  of  C  D  through 
which  the  curve  will  pas>. 

PROBLEM  26. — To  construct  an  hyperbola,  having  given  the 
distances  AO  and  a  O,  on  the  horizontal  axis,  from  the  centre  to 
either  vertex,  and  from  the  centre  to  either  focus. 

In  the  hyperbola,  the  difference  of  the  distances  of  any  point 
from  the  foci  is  equal  to  the  distance  between  the  vertices,  as  in 
the  ellipse  the  sum.  Lay  off  from  O  the  equal  distances  A  O  and 
<  >  B  to  the  vertices,  and  the  equal  distances  O  a  and  O  b  to  the 


17 

foci.  To  obtain  any  point  of  the  curve,  take  any  point  on  the  axis 
as  c;  with  c  A  and  c  B  as  radii,  and  a  and  b  as  centres,  descri1>e 
four  pair  of  intersecting  arcs,  as  in  the  ellipse ;  the  points  of  inter- 
section will  be  points  of  the  hyperbola.  By  taking  other  points 
on  the  axis,  other  points  of  the  curve  will  be  obtained  in  the  same 
manner. 

PROBLEM  27. — To  construct  a  curve  similar  to  a  given  curve  B 
A  C,  and  reduced  in  a  given  ratio,  one-half  for  instance. 

Draw  some  indefinite  line,  a  centre-line,  if  possible,  in  the  given 
curve,  as  A  D.  On  A  D,  lay  off  a  number  of  distances ;  at  the 
points  of  division,  erect  perpendiculars  to  meet  the  curve  above 
and  below  A  D.  Draw  an  indefinite  line  a  d,  and  on  it  lay  off  dis- 
tances bearing  respectively  to  those  laid  off  on  A  D,  the  given 
ratio.  Through  these  points  of  division  draw  perpendiculars,  and 
lay  off  on  them  above  and  below  a  d,  distances  bearing  the  given 
ratio  to  those  on  the  perpendiculars  to  A  D. 

PROBLEM  28. — To  describe  a  given  number  of  circles,  six,  for 
instance,  within  a  given  circle  O,  tangent  to  each  other  and  to  the 
given  circle. 

Divide  the  circumference  of  the  given  circle  into  twice  as  many 
parts  as  the  number  of  circles  to  be  describe,  1  2,  2  3,  etc.  To  ob- 
tain the  first  circle,  draw  a  tangent  to  the  given  circle  at  1,  la; 
produce  O  2  1  a,  at  a.  Lay  off  on  O  a,  from  a  inwards  the  distance 
1  a  to  b,  since  tangents  to  a  circle  are  of  equal  length.  At  b  draw 
a  perpendicular  to  Oa,  meeting  O  1  in  c.  With  c  as  a  centre  and  c 
1  as  a  radius,  describe  a  circle.  From  O  as  a  centre  and  O  c  as  a 
radius,  describe  a  circle  intersecting  the  alternate  radii  O  3,  O  5, 
etc..  in  points  which  will  be  centres  of  the  required  circles. 

PROBLEM  29. — To  construct  a  mean  proportional  to  two  giv- 
en lines  A  D  and  D  B. 

With  the  sum  of  these  lines  as  a  diameter,  describe  a  semicircle 
A  C  B.  At  the  point  D,  between  the  two  lines,  erect  a  perpendic- 
ular, meeting  the  circumference  at  C.  DC  will  be  the  mean  pro- 
portional required. 

PROBLEM  30. — To  divide  a  line  a  b  into  the  same  proportional 
parts  as  a  given  line  A  B  is  divided  by  the  point  C.  Draw  a  b  par- 
allel to  A  B,  and  draw  lines  through  A  and  a,  B  and  b,  till  they 
meet  in  d.  Draw  O  d,  the  point  c  will  divide  a  b  into  the  same 
proportional  parts  as  C  divides  A  B. 


18 


PROJECTIONS. 


If  we  wish  to  represent  a  solid  body  by  drawing,  and,  at  the 
same  time,  to  show  the  true  dimensions  of  that  body,  we  must 
have  two  or  more  views,  or  projections,  of  it  on  as  many  different 
planes.  Take  for  example  a  cube.  In  order  to  show  it  in  a  draw- 
ing, we  must  have  views  of  more  than  one  face,  in  order  to  show 
that  the  body  has  three  dimensions.  We  will  consider  the  cube  to 
be  behind  one  plate  of  glass  and  below  another,  and  in  such  a  posi- 
tion that  two  of  its  faces  are  parallel  to  these  plates,  which  are 
respectively  vertical  and  horizontal.  Now  suppose  that  perpen- 
diculars are  dropped  from  ever\-  corner  of  the  cube  to  each  of  these 
plates.  The  points  where  these  perpendiculars  pierce  the  surface 
of  the  plates,  are  called,  respectively,  the  vertical  and  horizontal 
projections  of  the  corners  of  the  cube.  If  these  points  be  joined 
bv  lines,  corresponding  to  the  edges  of  the  cube,  we  shall  have  in 
this  case,  exact  figures  of  the  two  faces  of  the  cube  that  are  paral- 
lel to  these  plates.  These  two  figures  are  called,  respectively, 
the  vertical  and  horizontal  projections  of  the  cube,  accord- 
ing as  they  are  on  the  vertical  or  horizontal  plates.  In  this 
way  we  may  get  two  views  of  any  solid  object,  supposing  it  to  be 
in  such  a  position  as  that  of  the  cube,  in  the  case  just  noticed, 
with  reference  to  two  plates  of  glass,  which  we  will  now  call  the 
vertical  and  horizontal  planes  of  projection. 

If  the  object  has  a  third  side  very  different  from  the  two  shown 
in  this  way,  we  may  consider  it  to  Tie  projected  on  a  third  plane 
perpendicular  to  the  two  others,  and  on  the  side  of  the  object  to 
be  represented. 

A  fourth  side  may  in  the  same  way  be  represented  on  a  fourth 
plane;  but  three  projections  are  generally  all  that  are  needed  to 
show  even  very  complicated  objects ;  and  in  most  cases  two  pro- 
jections, a  vertical  and  a  horizontal,  are  all  that  are  necessarv, 
lines  on  the  opposite  faces  being  shown  by  dotted  lines  on  the  pro- 
jections of  the  faces  toward  the  planes.* 


•  We  have  considered  the  planes  of  projection  to  be  in  front  and 
above  the  object  to  to  be  represented,  but  drawings  are  often  made  with 
the  planes  behind  and  below.  It  is.  however,  believed  that  the  method 
given  is  better  for  practical  use.  Details  are  often  shown  as  projected  on 
oblique  planes,  as  planes  parallel  or  at  right  angles  to  the  axis  of  an  ob- 
ject. 


19 

As  it  is  not  convenient  to  have  two  or  more  separate  draw- 
ings of  an  object  on  different  planes,  as  would  be  necessary  if  we 
were  to  represent  the  projections  of  the  body  in  their  true  posi- 
tions; we  may  consider  that  the  body  has  been  projected  in  the 
manner  indicated,  and  that  the  planes  of  projection  have  been 
revolved  about  their  interesctions  so  as  to  bring  them  all  into  the 
horizontal  plane,  with  the  end  views,  if  any,  on  the  right  and  left 
of  the  vertical  projections,  and  the  horizontal  projection  above 
the  vertical.  In  this  way  we  may  bring  all  the  different  views  or 
projections  into  the  plane  of  the  top  surface  of  the  drawing  paper ; 
and  by  representing  the  intersections  of  the  planes  of  projections 
by  lines,  we  may  show  all  the  projections  in  their  true  relative 
position  in  one  drawing. 

The  line  that  represents  the  intersection  of  the  vertical  and 
horizontal  planes  of  projection,  is  called  the  ground  line.  The 
ground  line,  as  well  as  the  other  lines  of  intersection  of  the  planes 
of  projection,  is  often  omitted  in  actual  drawings. 

It  will  appear  on  consideration  of  the  method  of  projection 
that  the  distances  of  the  projections  of  any  point  from  the  ground 
line  show  the  true  position  of  the  point  in  space,  with  reference  to 
the  planes  of  projection.  Suppose,  for  example,  the  horizontal 
projection  of  a  point  to  be  one  inch  above  the  ground  line,  and 
the  vertical  projections  to  be  two  inches  below  the  same  line,  this 
shows  that  the  true  position  of  the  point  in  space  is  one  inch  back 
of  the  vertical  plane  and  two  inches  below  the  horizontal  plane. 
Moreover,  it  may  easily  be  demonstrated  that  the  two  projections 
of  a  point  always  lie  in  a  common  perpendicular  to  the  ground 
line. 

As  lines  are  determined  by  locating  points  in  them,  the  princi- 
ples just  given  appljr  in  getting  the  projections  of  any  figure  that 
can  be  represented  by  lines.  In  the  problems  in  projection,  follow- 
ing, the  ground  lines  must  be  drawn  and  the  points  located  in  the 
manner  just  indicated. 

An  object  may  be  in  any  position  whatever  with  reference  to 
the  planes  of  projection ;  but  for  convenience  the  body  is  usually 
considered  to  be  in  such  a  position  that  the  vertical  projection  will 
show  the  most  important  view  of  the  object,  such,  for  example,  as 
the  front  of  a  building. 

In  representing  an  object  of  this  kind  in  projection,  the  front 
of  the  object  is  usually  considered  parallel  to  the  vertical  plane  of 
projection. 

The  vertical  projection  of  an  object  is  called  its  elevation,  and 
the  horizontal  projection,  its  plan.  The  other  projections  are 
called  end  views,  or  sections,  according  as  they  represent  an  end 
or  some  part  cut  by  a  plane  passing  through  the  object. 


20 

Hy  the  methods  of  projection  just  explained  each  projection 
represents  the  view  of  the  object  a  person  would  have  were  the 
eye  placed  on  the  side  of  the  object  represented  by  the  projection 
and  at  an  infinite  distance  from  it.  When  an  object  is  viewed  from 
a  finite  distance  it  is  seen  in  j>erspective  and  not  as  it  really  is. 
Projections  show  an  object  as  it  really  is,  and  not  as  it  appears  in 
perspective.  Projections  are  therefore  used  to  represent  bodies  in 
their,  true  form  and  are  employed  as  working  drawings,  in  which 
a  body  to  be  constructed  is  represented  as  it  would  appear  in 
projection  when  finished. 

LINE   SHADING. 

In  order  to  give  the  projection  of  a  bod\-  the  appearance  of 
relief  the  light  is  supposed  to  come  from  some  particular  direction, 
and  all  lines  that  separate  light  faces  from  dark  ones  are  made 
heavy. 

The  direction  of  the  light  is  generally  taken  for  convenience  at 
an  angle  of  45  degrees  from  over  the  left  shoulder  as  the  person 
would  stand  in  viewing  the  projections,  or  in  making  the  draw- 
ing ;  and  in  all  cases  the  projections  of  this  book  are  to  be  shaded 
with  the  light  so  taken. 

It  will  be  readily  seen  on  considering  the  direction  of  the  light, 
that  the  elevation  of  a  solid  rectangular  object  with  plane  faces 
in  the  common  position,  will  have  heavy  lines  at  the  lower  and 
right  hand  sides,  and  that  the  plan  will  have  heavy  lines  on  the 
upper  and  right  hand  sides. 

When  a  body  is  in  an  oblique  position  with  reference  to  the 
planes  of  projection,  the  heavy  lines  of  the  projections  may  be 
determined  by  using  the  forty -five  degree  triangle  on  the  T-square. 
If  we  apply  this  triangle  to  the  T-square,  so  that  one  of  its  edges 
inclines  to  the  T-square  at  an  angle  of  forty -five  degrees  upwards, 
and  to  the  right  this  edge  will  represent  the  horizontal  projection 
of  a  ray  of  light:  and  by  noticing  what  lines  in  the  plan 
of  the  object  this  line  crosses,  it  may  readily  lie  seen  what 
faces  in  the  deration  will  receive  the  light  and  what  faces  will  be 
in  the  shade.  By  applying  the  triangle  so  that  an  edge  will 
make  an  angle  of  forty-five  degrees  downward  and  to  the 
right,  this  edge  will  represent  the  vertical  projection  of  a  ray  of 
light,  and  by  applying  it  to  the  elevation,  the  faces  in  the  plan 
that  will  IK:  in  light  and  shade  may  lie  determined. 

Where  the  limiting  line  of  projection  is  an  element  of 
a  curved  surface  as  in  the  elevation  of  a  vertical  cylinder,  that 
line  should  not  be  shaded.  The  plan  of  the  vertical  cylinder, 
which  is  a  circle,  should  be  shaded,  for  the  circumference  is  an  edge 
separating  light  from  dark  portions  of  the  object.  In  this  case 


21 

the  darkest  shade  of  the  line  should  be  where  the  diameter,  that 
makes  an  angle  of  forty-five  degrees  to  the  right  with  the 
T-square  cuts  the  circumference  above  and  the  lightest  part  should 
be  where  this  diameter  cuts  the  circumference  below.  The  dark 
part  should  taper  gradually  into  the  light  part. 

Lines  that  separate  parts  of  a  body  that  are  flush  with  each 
other,  as  in  joints,  should  never  be  shaded,  and  when  a  line  that 
would  otherwise  be  shaded,  rests  on  a  horizontal  plane  as  in  the 
first  positions  of  the  following  problem,  it  should  not  be  shaded. 

The  shaded  lines  of  a  projection  need  not  be  very  heavy  if 
the  light  lines  are  made,  as  they  should  be,  as  fine  as  possible. 

THE   PLATE   OF   PROJECTIONS. 

The  plate  of  projections  is  to  be  of  the  same  size  as  those  of 
geometrical  problems,  and  will  contain  twelve  problems  in  the 
position  shown  in  the  cut.  The  border  should  be  drawn  first, 
leaving  a  margin  of  one  and  a  quarter  inches  at  the  top  as  in  the 
previous  plates.  Divide  next  the  space  within  the  border  line  in- 
to twelve  equal  rectangles,  by  drawing  five  vertical  and  one  hori- 
zontal lines.  Draw  a  ground  line  in  each  rectangle  two  inches  and 
a  half  below  the  top,  making  the  ends  of  the  ground  lines  within 
a  quarter  of  an  inch  of  the  vertical  lines  dividing  the  space.  At  a 
distance  of  two  inches  and  three-quarters  below  the  ground  line 
draw  a  broken  horizontal  line,  as  shown  in  the  plate. 

The  objects  projected  in  the  following  problems  are  all  sup- 
posed to  rest  on  a  horizontal  plane  below  the  horizontal  plane  of 
projection  and  behind  the  vertical  plane  of  projection.  The  ground 
line,  as  has  already  been  noticed,  represents  the  intersection  of 
the  two  planes  of  projection  before  the  vertical  plane  is  revolved 
into  the  plane  of  the  horizontal.  The  broken  line  below  represents 
the  intersection  of  the  vertical  plane  with  the  plane  on  which  the 
body  rests. 

In  order  to  get  a  good  conception  of  the  position  of  the  object 
suppose  that  the  body  rests  on  the  drawing  table,  and  that  a 
plate  glass  be  held  above  and  parallel  to  the  table,  and  another 
plate  be  held  in  front  and  vertical.  The  position  of  the  object  in 
relation  to  the  planes  represented  by  the  plates  of  glass  will  be  the 
same  as  that  of  the  cube,  which  we  considered  in  explaining  pro- 
jections in  general. 

The  ground  line  and  the  broken  line  below  will  represent  in 
this  case,  respectively,  the  intersections  of  the  plates  of  glass  with 
each  other  and  with  the  top  surface  of  the  drawing  table.  A  sheet 
of  paper  may  now  be  put  in  place  of  the  horizontal  plate  glass, 
and  it  will  represent  the  revolved  position  of  the  planes  precisely 
as  they  are  in  the  drawing. 


22 

In  the  descriptions  of  the  problems,  G  L  refers  to  the  ground 
line,  or  intersection  of  the  planes  of  projection,  and  G'  L'  refers  to 
the  line  of  intersection  of  the  vertical  plane  of  projection  with  the 
plane  on  which  the  body  rests. 

The  plate  must  be  lettered  to  show  the  general  title,  and  the 
number  of  problems,  as  shown  in  Plate  III.  The  letters  used  in 
describing  the  problems,  however,  need  not  be  drawn. 

All  lines  that  would  not  be  seen  from  the  position  indicated  by 
the  projection  in  question,  must  be  indicated  by  fine  dots.  The 
problems  in  the  finished  plate  must  be  shaded  according  to  the  di- 
rections above. 

PROBLEMS  IN  PROJECTION. 

PROBLEM  1. — To  construct  the  projections  of  a  prism  one  and 
a  quarter  inches  square  at  the  base  and  two  and  a  quarter  inches 
in  height,  of  whose  laces  one  rests  on  a  horizontal  plane,  and  one 
is  parallel  to  the  vertical  plane  of  projection.  Draw  the  square, 
A  B  C  D,  equal  to  the  top  face  of  the  prism,  above  G  L,  with  C  D 
one-quarter  of  an  inch  from  G  L  and  parallel  to  it.  Draw  from  C 
and  D  lines  perpendicular  to  G  L,  and  prolong  them  lielow  until 
they  intersect  G'  L'.  Measure  off  the  height  of  the  prism  from  G' 
L',  and  draw  a  horizontal  line  for  the  top  line  of  the  elevation. 
The  rectangle,  E  F  G  H,  formed  below  the  line  last  drawn  will  be 
the  elevation  of  the  prism,  and  the  square  above  G  L  will  be  the 
plan.  No  dotted  lines  will  api>ear  in  this  problem,  as  all  the  lines 
on  the  opposite  sides  of  the  object  will  1)e  covered  by  the  full  lines 
in  front.  Shade  according  to  the  directions  above. 

PROBLEM  2. — To  revolve  the  prism  of  Problem  1  through  a 
given  angle  about  an  edge  through  H,  so  that  the  planes  parallel 
to  the  vertical  plane  shall  remain  so. 

Locate  H  on  G  L  as  far  to  the  right  in  its  rectangle,  as  H,  in 
Problem  1 ,  is  from  the  border  line.  As  the  revolution  has  been 
parallel  to  the  vertical  plane,  the  elevation  will  be  unchanged  in 
form  and  dimensions,  but  will  be  inclined  to  G  L.  Lay  off  G  H, 
making  the  given  angle  of  revolution  with  G'  L.'  Complete  E  F 
G  H,  on  G  H  as  base.  Since  the  body  has  revolved  parallel  to  the 
vertical  plane,  the  horizontal  projections  of  lines  perpendicular  to 
the  vertical  plane  as  A  C  and  B  D,  have  not  changed  in  length, 
but  those  of  lines  parallel  to  the  vertical  plane,  as  A  B  and  C  D, 
will  be  shortened. 

Consider  these  facts,  and  that  the  two  projections  of  a  point 
are  in  the  same  line  perpendicular  to  G  L,  the  following  is  seen  to 
be  the  construction  of  the  plan  : 

Prolong  A  B  and  C  D  of  Problem  1,  indefinitely  to  the  right. 
As  E  F  G  H  represent  the  same  points  in  both  elevations,  erect 


• 

M 


100 

65 


CVJ.       Or 


23 

perpendiculars  from  each  of  these  points  in  Problem  2,  intersecting 
the  indefinite  lines  just  drawn,  for  the  plans  of  the  same  points. 
A  B  C  D  and  I  J  K  L  will  be  the  bases  of  the  prism  in  its  revolved 
position.  K  L  is  to  be  dotted  because  not  seen.  Shade  as  directed 
above.* 

PROBLEM  3. — To  revolve  the  prism,  as  seen  in  its  last  position, 
through  a  horizontal  angle,  that  is  about  a  line  through  H,  per- 
pendicular to  the  horizontal  plane. 

As  the  revolution  is  parallel  to  the  horizontal  plane,  the  plan 
is  changed  only  in  position,  not  in  form  or  dimensions. 

Therefore,  draw  the  plan  of  Problem  2  inclined  to  G  L  at  the 
angle  of  revolution,  taking  L  in  L  D  produced  from  Problem  2. 

Now  as  each  point  of  the  body  revolves  in  a  horizontal  plane, 
its  vertical  projection  will  move  in  a  straight  line  parallel  to  G  L. 
Hence  we  make  the  following  construction  for  the  elevation:  In 
the  case  of  any  point,  as  B,  in  the  plan,  draw  a  perpendicular 
from  this  point  to  G  L,  and  from  F,  which  is  the  elevation  of  B  in 
Problem  2,  draw  an  indefinite  line  parallel  to  G.  L.  The  intersec- 
tion of  these  lines  gives  the  elevation  of  the  point  in  its  revolved 
position.  Proceed  in  the  same  way  with  all  the  other  points. 

It  will  be  noticed  that  in  the  three  positions  of  the  body  just 
taken  the  plan  is  drawn  first  in  Problem  1,  the  elevation  first  in 
Problem  2,  and  the  plan  first  in  Problem  3.  The  reasons  for  so 
proceeding  are  evident  from  the  constructions. 

This  order  will  hold  true  in  all  the  problems  following. 

PROBLEM  4. — To  construct  the  projections  of  a  regular  hexa- 
gonal pyramid,  and  the  projections  of  a  section  of  that  pyramid 
made  by  a  plane  which  is  perpendicular  to  the  vertical  plane. 

The  height  of  the  pyramid  is  to  be  the  same  as  that  of  the 
prism  in  Problem  1,  and  the  diameter  ol  the  circumscribing  circle 
of  the  base  is  to  be  two  inches. 

The  part  of  the  pyramid  above  the  section  is  to  be  represented 
by  dotted  lines  and  the  lower  part,  or  frustum,  in  full  lines. 

To  find  the  projections  of  the  pyramid,  draw  a  regular  hex- 
agon, A  B  C  D  E  P,  above  G  L,  with  the  lines  joining  the  opposite 
vertices  for  the  plan  of  the  p3'ramid ;  draw  perpendiculars  from 
the  vertices  G  L.  The  intersections  of  these  perpendiculars  with 
G  L  will  be  the  elevations  of  the  corners  of  the  base.  Erect  a  per- 
pendicular from  the  center  of  the  elevation  of  the  base,  and  on  it 


•Nothing  more  will  be  said  about  shading,  but  it  is  to  be  understood 
that  each  projection  is  to  be  shaded,  with  the  light  taken  from  over  the 
left  shoulder  at  an  angle  of  forty-five  degress  with  the  horizontal  plane. 
The  shading  is  a  very  important  point  in  the  problems. 


24 

measure  off  the  vertical  height  to  H,  join  Hwith  the  points  on  the 
base  for  the  elevations  of  the  edges. 

To  draw  the  projections  of  the  section,  assume  L  K  making 
an  angle  with  G  L,and  cutting  the  pyramid  as  shown  in  the  plate. 
Draw  M  N  perpendicular  to  G  L,  and  vertically  above  L.  K  L 
represents  the  intersection  of  the  cutting  plane  with  the  vertical 
plane;  and  M  N,  its  intersection,  with  the  horizontal  plane. 

The  lines  are  called  the  traces  of  the  cutting  plane,  and  must 
be  represented  by  broken  lines  like  G  L.  The  elevation  of  the  sec- 
tion will  be  the  part  of  K  L  included  between  the  limiting  edges  of 
the  pyramid,  and  is  to  be  shown  a  lull  line. 

The  plan  of  the  section  is  found  by  erecting  perpendiculars  to 
G  L  from  the  points  where  K  L  cuts  the  elevations  of  the  edges  of 
the  pyramid,  and  by  finding  where  these  perpendiculars  intersect 
the  plans  of  the  same  edges.  T  represents  the  plan  and  elevation 
of  one  point  in  the  section. 

PROBLEM  5. — To  revolve  the  frustum  of  the  pyramid  in  Prob- 
lem 4  through  a  vertical  angle  about  J. 

Draw  S  I  equal  to  the  same  line  in  Problem  4,  and  making  a 
given  angle  with  G  L.  Construct,  on  S  I,  an  elevation  like  the 
one  in  Problem  4. 

Each  point  in  the  plan  may  be  found,  as  in  the  second  poblem 
of  the  prisms,  by  erecting  perpendiculars  from  the  points  on  the 
elevation,  and  finding  their  intersections  with  horizontals  drawn 
from  the  plans  of  the  same  points  in  Problem  4.  T  shows  the  plan 
and  elevation  of  the  point  T  in  Problem  4. 

PROBLEM  6. — To  revolve  the  frustum  of  Problem  5  through  a 
given  horizontal  angle. 

Draw  the  plan  like  that  of  Problem  5,  but  making  the  given 
angle  of  revolution  with  G  L. 

Each  point  in  the  elevation  may  be  found  by  drawing  perpen- 
diculars and  horizontals  respectively,  from  points  in  the  plan,  and 
trom  corresponding  points  in  the  elevation  of  Problem  5.  The 
revolution  of  the  point  T  is  indicated  by  the  dotted  lines. 

PROBLEM  7. — To  construct  the  projections  of  a  regular  octa- 
gonal prism,  one  of  whose  bases  is  in  a  horizontal  plane. 

Lay  out  a  regular  octagon,  one  inch  and  three-quarters  be- 
tween parallel  sides,  for  the  plan  ;  and  from  its  vertices  draw  ver- 
ticals to  G  L,  and  produce  them  below  G  L,  until  they  intersect 
G  L,  for  the  vertical  edges  of  the  prism.  Makefile  top  line  of  the 
elevation  three-eighths  of  an  inch  below  G  L. 

PROBLEM  8. — To  revolve  the  prism  of  Problem  7,  parallel  to 
the  vertical  plane,  through  a  given  angle. 


25 

Construct,  as  usual,  the  vertical  projection,  differing  only  in 
position  from  that  in  Problem  7.  In  the  case  of  any  point  as 
T,  to  find  its  plan,  erect  a  perpendicular  to  G  L  from  the  elevation 
of  the  point  and  draw  a  horizontal  from  the  corresponding  point 
in  the  plan  of  Problem  7.  The  bases  of  the  prism  in  this  position 
will  be  equal  octagons  though  not  regular. 

PROBLEM  9. — To  revolve  the  prism  of  Problem  8,  parallel  to 
the  vertical  plane,  through  a  given  angle. 

Draw  the  plan  like  that  in  Problem  8,  making  the  given  angle 
of  revolution  with  G  L.  In  the  case  of  each  point,  to  obtain  the 
elevation,  drop  a  vertical  from  the  plan  of  the  point,  and  draw  a 
horizontal  from  the  corresponding  point  in  the  elevation  of  Prob- 
lem 8.  The  bases  of  the  elevation  will  be  equal  octagons,  in 
which  the  parallel  sides  are  equal. 

The  point  T  is  the  same  point  that  is  marked  in  Problems  7 
and  8. 

PROBLEM  11. — To  construct  the  projections  of  the  frustum  of 
a  right  cone,  whose  base  is  in  a  horizontal  plane. 

Draw  a  circle  of  a  radius  of  seven-eighths  of  an  inch  for 
the  plan  of  the  base.  Drop  verticals  from  the  right  and  lefthand 
limits  of  this  circle  intersecting  G'L/,  for  the  elevation  of  the  base. 
Drop  another  vertical  from  the  center  of  the  circle,  and  on  it 
measure  the  same  length  from  G'  L/  as  that  of  the  prism  of  Prob.  7 
above.  This  will  give  the  elevation  of  the  apex  of  the  complete 
cone,  the  plan  of  which  is  the  center  of  the  circle  above.  Join  this 
point  with  the  two  ends  of  the  elevation  of  the  base  for  the  limit- 
ing elements  in  the  elevation. 

The  upper  base  of  the  frustum  in  this  case,  is  formed  by  a 
plane,  cutting  the  cylinder,  perpendicular  to  the  vertical  plane  of 
projection,  and  making  an  angle  with  the  horizontal  plane.  The 
cutting  plane  is  given  by  its  trace  on  the  plate.  This  upper  base 
will  be  an  ellipse,  as  is  every  section  of  cone  made  by  a  plane  that 
does  not  cut  the  base  of  the  cone. 

The  elevation  of  this  upper  base  will  evidently  be  that  part  of 
the  vertical  trace  of  the  cutting  plane  included  between  the  limit- 
ing elements. 

To  get  the  plan  of  this  base,  proceed  as  follows:  Divide  the 
straight  line,  representing  the  elevation  of  this  base,  into  any 
number  of  equal  parts,  and  through  these  points  of  division  draw 
horizontals  as  shown  on  the  plate. 

The  distances  of  these  points  from  the  axis  of  the  cone  are  evi- 
dently equal  to  the  lengths  of  the  horizontals  drawn  through 
these  points  in  the  elevation,  and  limited  by  the  axis  and  the  limit- 
ing elements.  Hence,  to  get  any  point,  like  A,  in  the  plan,  erect  a 


26 

vertical  from  the  elevation  of  that  point ;  and  with  O  the  plan  of 
any  point  in  the  axis,  as  a  centre,  and  the  horizontal  M  X  through 
the  point  in  the  elevation  as  a  radius,  describe  arcs  intersecting 
the  vertical.  A  and  the  point  above  it,  are  both  found  by  using  the 
same  radius,  getting  the  intersections  above  and  below  with  the 
vertical  from  the  point  in  the  elevation. 

PROBLEM  11. — To  revolve  the  frustum  of  Problem  10  through 
a  given  angle  parallel  to  the  vertical  plane. 

Construct,  as  in  all  the  preceding  similar  cases,  the  elevation 
of  the  preceding  problem,  making  the  given  angle  with  G  L. 

Get  the  plan  of  the  lower  base  by  locating  a  sufficient  num- 
ber of  points  in  the  same  way  that  the  corners  of  the  polygon  are 
located  in  Problem  5. 

To  get  the  plan  of  the  upper  base,  erect  verticals  from  points 
in  the  elevation  corresponding  to  those  marked  in  Problem  10, 
and  find  the  intersections  of  horizontals  from  the  plans  of  the 
same  points  in  Problem  10. 

In  case  the  frustum  were  turned  through  a  larger  angle  than 
that  shown  in  the  plate,  limiting  elements  would  show  tangent 
to  the  two  ellipses  in  the  plan. 

PROBLEM  12. — To  revolve  the  frustum  of  Problem  11  through 
a  given  horizontal  angle. 

Construct  the  plan  of  Problem  11;  making  the  given  angle 
with  G  L,  in  a  manner  similar  to  that  employed  in  .constructing 
the  plan  of  pyramid  in  Problem  9. 

Get  the  points  in  the  two  bases  b\"  dropping  verticals  from  the 
points  in  the  plan,  and  finding  the  intersections  of  horizontals 
from  corresponding  points  in  the  elevation  of  Problem  11.  Draw 
the  limiting  tangent  to  the  ellipses  found  in  this  way. 

The  orthographic  projections  of  an  ellipse  are  always  ellipses, 
the  circle  and  the  straight  line  being  special  cases  of  ellipses. 

PRACTICAL  APPLICATIONS  OF  PROJECTIONS. 
Roof  Truss. — ''Scale  one  half  an  inch  to  the  foot. 

In  this  case,  the  two  sides  being  symmetrical,  an  elevation  of 
one  half  and  a  section  through  A  B,  will  show  every  part,  and 
are  therefore  chosen  as  the  best  views  for  the  working  drawing. 

First,  lay  off  a  dotted   horizontal  line  B  C,  which  is  t wen ty- 

*The  scale  of  a  drawing  is  the  ratio  that  the  lines  on  the  drawing  bear 
to  the  actual  lengths  of  the  lines  on  the  object.  The  scale  should  always 
be  stated  on  a  drawing;  and  may  be  given  as  a  fraction  like  \y,  1/4,  etc., 
or  it  may  be  stated  as  a  certain  number  of  inches  to  the  foot. 


..n 


. 


27 

five  feet  in  length.  Then  at  E  erect  the  vertical  D  E  making  E  B 
eight  feet  and  E  D  nine  feet,  reducing  eacli  to  the  proper  scale. 

Join  C  and  D.  These  lines  are  the  center  lines  of  the  principal 
timbers  in  the  truss  and  of  the  iron  rod  D  E. 

Draw  next  the  timber  of  which  C  B  is  the  centre,  fifteen  inches 
deep,  leaving  the  end  near  C  unfinished  until  the  rafter  is  drawn. 
The  other  two  timbers  of  which  the  centre  lines  have  been  drawn 
are  twelve  inches  deep.  Draw  the  lines  parallel  to  the  center  lines. 
The  line  forming  the  joint  at  D  is  found  by  joining  the  intersec- 
tions of  the  lines  of  the  timbers.  The  joint  above  C  is  formed  by 
cutting  in  three  inches  in  a  direction  perpendicular  to  the  upper 
edge  of  C  D  and  joining  the  end  of  this  perpendicular  to  the  inter- 
section of  the  lower  line  of  C  D  \vith  the  upper  line  of  C  B. 

The  hatched  pieces  near  D,  H  and  C,  are  sections  of  the  pur- 
lins, long  pieces  resting  on  the  truss  and  supporting  the  rafters. 
These  purlins  are  cut  into  the  rafters  and  into  the  truss  one  inch, 
with  the  exception  that  the  one  near  C  is  not  cut  into  the  truss. 
These  should  next  be  drawn,  rectangles  ten  inches  by  six  inches. 
The  center  line  of  the  one  near  C  is  a  prolongation  of  the  short 
line  of  the  joint  at  C,  the  one  near  D  is  three  inches  below  the  joint 
at  the  top,  and  the  third  is  half  way  between  the  other  two. 

The  center  line  of  H  E  should  start  from  the  lower  part  of  the 
center  line  of  the  middle  purlin,  and  the  top  edge  should  meet  the 
top  edge  of  C  B  in  D  E.  With  one  half  the  depth  of  H  E,  four 
inches,  as  a  radius,  describe  an  arc  with  the  point  where  D  E 
meets  the  upper  line  of  C  B  as  a  center.  Draw  the  center  line 
through  the  point  indicated  above,  and  tangent  to  this  arc.  Cut 
in  at  E  one  third  the  depth  of  H  E,  or  two  and  two-thirds  inches 
and,  at  H,  one  inch,  in  the  way  shown  in  the  plate. 

Draw  next  the  rafter,  F  A,  twelve  inches  deep  and  eight  inches 
from  the  top  of  C  D.  Cut  into  the  rafter  a  horizontal  distance  of 
six  inches  for  the  end  of  the  beam  C  B,  and  make  the  end  of  the 
rafter  in  line  with  the  bottom  of  C  B. 

Make  a  short  end  of  the  rafter  on  the  other  side  as  shown  at 
A,  and  show  the  horizontal  pieces  broken  off  as  shown  in  the 
plate.  Make  the  rod,  D  E,  an  inch  and  a  half  in  diameter  with 
washer  and  nut  at  the  lower  end,  and  with  a  head  and  an  angle 
plate  running  to  the  purlin  at  the  upper.  Make  the  washer  and 
angle  plate  one  inch  thick,  the  washer  four  inches  in  diameter  and 
the  nut  and  head  according  to  the  standard.  The  short  bolt  near 
E  is  one  inch  in  diameter  and  the  head,  nut  and  washer  of  two- 
thirds  the  size  those  on  in  D  E.  The  short  bolt  near  C  is  precisely 
like  the  one  just  described,  make  the  angular  washer  at  the  bottom 
of  the  same  diameter  and  at  right  angles  to  the  bolts  as  in  the 
other  washers. 


28 

The  sectional  view  at  the  right  is  formed  by  projecting  lines 
across  from  the  elevation  just  drawn  and  measuring  off  the  prop- 
er widths.  The  timbers  of  the  truss  proper  are  all  twelve  inches 
wide,  the  rafter  is  three  inches  wide  and  the  purlins  are  broken  off 
so  as  to  show  about  two  feet  in  length  of  each.  » 

Great  care  must  be  taken  in  inking  not  to  cross  lines,  and 
those  nearest  the  observer  in  any  view  should  be  full  lines,  and 
those  hidden  by  them  should  be  broken  off  or  dotted. 

Hatching,  as  it  is  called,  is  a  method  of  representing  a  surface 
cut,  as  in  a  section,  and  it  is  done  by  drawing  fine  parallel  lines 
at  an  angle  of  forty-five  degrees  with  the  vertical  and  very  near 
together. 

In  case  two  different  pieces  joining  are  cut,  the  lines,  should  be 
at  right  angles  to  distinguish  the  two  sections.  There  are  no 
such  cases  in  this  example  however.  It  is  very  important  that  a 
hatched  surface  shall  look  even,  and  this  can  only  be  effected  by 
making  all  lines  of  the  same  width,  and  the  same  distance  apart. 

The  title,  which  is  given  in  italics  at  the  beginning  of  this  de- 
scription, may  be  placed  within  the  border  as  indicated. 

The  student  will  be  allowed  to  choose  a.ny  mechanical  letters 
for  the  title,  but  the  heights  rmist  be  three-eighths  of  an  inch  in  the 
words  Roof  Truss,  which  must  be  in  capitals,  and  the  letters  in  the 
words  indicating  the  scale  are  to  be  one-half  as  high.  The  scale 
should  be  put  on  thus:  Scale  Vi"  ~  1',  one  dash  indicating  feet 
and  two  dashes  inches. 

Stub  End  of  a  Connecting  Rod. 
(One-half  Size.) 

The  projections  chosen  to  represent  this  object  are  a  front  ele- 
vation, and  a  side  elevation.  In  this  case  these  projections 
show  the  different  parts  much  more  clearly  than  the}'  could  be 
shown  in  plan  and  elevation. 

The  shaded  portion  on  the  front  elevation  shows  what  would 
be  seen  if  the  brasses  were  cut  away  on  the  line  D  E  of  the  side  ele- 
vation, this  is  done  to  show  the  lining  of  babbit  or  white  metal. 

This  example  illustrates  the  necessity  of  hatching  to  distin- 
guish the  cut  portions  from  those  beyond.  It  also  shows  the 
proper  method  of  representing  the  different  pieces,  shown  in  sec- 
tion, by  lines  running  in  different  directions  on  adjacent  pieces. 

The  dimensions  to  be  used  on  the  full  sized  drawing  are  marked 
in  inches  on  the  cut.  Remember  that  the  drawing  is  to  be  one- 
half  size.  The  arrowheads  on  either  side  of  the  dimension  marked 


LJ 
I- 


J 

d 


29 

represent  the  limits  of  the  dimension.  It  will  be  noticed  that 
some  of  the  dimensions  at  the  top  are  diameters,  whilst  others 
are  radii. 

Divide  the  space  within  the  border  on  a  half-sheet  of  Imperial 
by  two  vertical  lines  making  three  equal  spaces.  Use  these  lines 
as  centre  lines  of  the  two  projections.  Draw  the  front  elevation 
first.  Commence  by  assuming  on  the  line  A  B,  the  centre  C  of  the 
inner  circle,  near  the  top,  and  describe  the  inner  circle,  about  this 
center  with  the  given  radius.  The  center  of  this  circle  should  be 
taken  far  enough  below  the  border  at  the  top  to  leave  about  the 
same  amount  of  margin  above  and  below  when  the  elevation  is 
completed. 

The  second  pair  of  half  circles  it  will  be  seen,  are  1/4"  at  top 
and  bottom  from  the  inner  circle  and  %"  at  either  side,  by  trial  a 
centre  for  the  upper  one  can  easily  be  found  on  the  line  A  B,  below 
C,  and  for  the  lower  one,  above  C.  All  other  arcs  are  drawn  with 
C  as  centre.  The  front  elevation  should  be  drawn  first,  as  most  of 
the  lines  of  the  side  elevation  are  obtained  by  projection  from  it. 

The  curves  at  the  bottom  of  each  elevation  are  arbitrary  and 
only  need  to  be  tangent  to  the  lower  lines  in  about  the  propor- 
tions shown.  The  horizontal  lines  in  the  side  elevation  may  be 
projected  from  corresponding  points  in  the  front  elevation.  All 
the  horizontal  distances  are  indicated  on  the  side  elevation,  but 
the  vertical  distances,  being  the  same  as  in  front  elevation,  are 
purposely  omitted.  The  most  difficult  part  of  the  work  in  this 
drawing  is  to  make  the  hatching  even.  Use  a  sharp  pen  and  make 
all  the  lines  of  the  same  width  and  the  same  distance  apart.  This 
drawing  should  be  line  shaded  according  to  previous  directions. 

It  will  be  seen  that  the  side  elevation  which  represents  the  left 
side  of  the  object  is  placed  at  the  right.  This  is  according  to  the 
method  of  projection  in  which  the  glass  or  vertical  plane  is  behind 
the  object  instead  of  in  front.  The  latter  is  generally  considered 
better. 

Projections  of  Screws. 

The  thread  of  a  screw  may  be  considered  to  be  generated  by 
a  section  moving  uniformly  around  a  cylinder,  and  at  the  same 
time  uniformly  in  a  direction  parallel  to  the  axis  of  the  cylinder. 
Plate  VI  shows  the  true  projections  of  a  V-threaded  screw  at  the 
left,  and  of  a  square  threaded  screw  at  the  right. 

V-TuKEADED  SCREW. — Commence  by  describing  a  semi-circle 
with  a  radius  of  one  inch  and  a  half,  as  shown  in  the  outer  dotted 
circle  in  the  plan.  This  will  l>e  the  half  plan  of  the  tfuter  part  of 
the  thread.  Drop  verticals  from  the  outer  limits  of  the  semi- 
circle for  the  limiting  lines  of  the  V  threads  in  the  elevation.  The 


30 

projections  of  the  head  and  an  outline  of  a  section  of  the  nut 
should  next  be  drawn.  The  standard  dimensions  of  the  heads  and 
nuts  are  expressed  by  the  following  formulae,  in  which  d  is  the  out- 
side diameter  of  the  screw,  h  the  thickness  of  the  head  or  nut  and 
D  the  distance  between  the  parallel  sides  of  the  head  or  nut: 
D= 


Construct  the  projections  of  the  heads  and  nuts  according  to 
this  standard,  and  show  the  hexagonal  head  finished  as  in  the 
plate.  The  short  arcs  that  cut  off  the  corners  are  described  with 
the  middle  of  the  lower  line  of  the  nut  as  a  center,  and  the  longer 
arcs  bounding  the  top  faces  of  the  head  are  described  with  the 
middle  point  of  the  lower  line  of  each  face  as  a  center.  The  top  of 
the  head  is  a  circle,  as  shown  in  the  plan. 

A  section  of  a  standard  V-thread  is  an  equilateral  triangle  all 
the  angles  of  which  are  sixty  degrees,  hence  the  outlines  of  the 
sides  of  the  elevation  may  be  drawn  by  means  of  the  thirty  de- 
gree triangle  used  on  the  T-square. 

Before  drawing  these  triangles,  however,  the  pitch  must  be  de- 
termined. The  pitch  of  a  screw  is  the  distance  from  any  point  on 
a  thread  to  another  point  on  the  same  thread  on  a  line  parallel  to 
the  axis.  The  pitch  is  usuall}'  expressed  by  stating  the  number  of 
threads  to  the  inch.  This  screw  has  two  threads  to  the  inch, 
therefore  the  pitch  is  one-half  an  inch.  This  i-epresents  the  ad- 
vance in  the  direction  of  the  axis  during  one  revolution. 

Lay  off,  then,  on  the  limiting  line  at  the  left,  distances  of  one- 
half  an  inch,  commencing  at  the  bottom  of  the  head. 

Through  these  points  draw  lines  as  indicated  above,  making  a 
series  of  triangles.  The  inner  intersections  of  these  lines  will  be  in 
a  vertical  line,  which,  projected  up,  gives  the  radius  of  the  inner 
dotted  semicircle  in  the  plan.  The  semi-circle  is  a  half  plan  of  the 
bases  of  the  threads.  As  the  thread  advances  a  distance  equal  to 
the  pitch  in  a  whole  revolution,  is  evident  that  in  a  half  revolution 
the  advance  will  be  equal  to  half  the  pitch;  therefore  commence  on 
the  right  hand  limiting  line  with  the  first  space  a  quarter  of  an 
inch,  and  from  this  point  on,  make  the  spaces  equal  to  the  pitch. 
Describes  a  series  of  triangles  on  this  side  in  the  same  way  as  be- 
fore. 

Every  point  in  the  generating  triangle  describes  a  helix  as  it 
revolves  about  and  at  the  same  time  moves  in  the  direction  of  the 
axis  of  the  screw.  It  is  evident  that  the  helices  described  by  the 
vertices  of  the  triangle  will  be  the  edges  of  the  intersections  of  the 
threads.  The  manner  of  getting  the  projections  of  these  lines  will 
be  described.  The  plans  of  these  helices  will  be  the  circles  which 
have  just  been  obtained  and  which  are  shown  in  the  plate  in  dotted 


31 

lines.  Draw  from  the  outer  vertex  of  one  of  the  triangles  repre- 
senting the  edges  of  the  threads,  an  indefinite  line  toward  the  left 
as  shown  in  the  plate.  Divide  the  semi-circle  above  into  a  number 
of  equal  spaces,  eight  at  least,  and  draw  radii  to  these  points  of 
division.  Lay  off  the  same  number  of  equal  divisions  on  the  in- 
definite line,  and  at  the  last  point  erect  a  perpendicular  equal  in 
length  to  one-half  the  pitch.  Join  the  end  of  this  line  with  the 
right  hand  end  of  the  horizontal  line,  forming  a  triangle.  Erect 
verticals  from  each  point  of  the  division  of  the  horizontal  line.  To 
find  any  point,  like  A,  in  the  helix  forming  the  edge  of  the  threads, 
drop  a  vertical  from  one  of  the  divisions  of  the  semi-circle,  and 
find  where  it  intersects  a  horizontal  drawn  from  a  corresponding 
point  on  the  diagonal  line  of  the  triangle  at  the  left,  counting 
the  same  number  of  spaces  from  the  right  on  the  diagonal  line  as 
the  point  taken  on  the  semi-circles  is  from  the  left.  As  many  points 
may  be  found  this  way  as  there  are  on  the  semi- circle.  Join  these 
points  by  using  the  irregular  curve. 

The  points  in  the  helices  at  the  bases  of  the  threads  may  be 
found  in  the  same  way  as  shown  by  the  dotted  lines,  the  equal 
divisions  of  the  semi-circle  in  this  case  being  where  the  radii  of  the 
center  circle  cut  this  one.  The  reason  of  this  construction  will  be 
plain  on  considering  that  the  equal  spaces  on  the  arc,  represent 
equal  angles  of  revolution  of  the  generating  triangle;  and  the  dis- 
tances between  the  horizontals  drawn  from  the  points  of  division 
of  the  diagonal  line,  represent  the  equal  rates  of  advance  in  the 
direction  of  the  axis. 

As  the  curves  at  the  edges  of  the  different  threads  are  all  alike, 
a  pattern  should  be  made,  from  thin  wood,  of  the  one  constructed, 
and  this  should  be  used  to  mark  all  the  long  curves  of  the  screw 
and  nut.  Another  curve  should  be  made  for  the  inner  helices. 

The  helices  will  evidently  be  continuous  from  one  end  of  the 
screw  to  the  other,  but  the  dotted  lines  which  would  show  the 
parts  on  the  back  side  are  left  out  in  order  that  the  drawing  may 
not  be  confused  by  too  many  lines. 

In  the  plate  the  screw  is  shown  as  entering  only  a  short  dis- 
tance into  the  nut  which  is  shown  in  section  below. 

The  threads  of  the  nut  are  the  exact  counter  parts  of  the 
threads  of  the  screw;  but  as  the  threads  on  the  back  side  of  the 
nut  are  shown  in  this  section,  the  curves  run  in  the  opposite  direc- 
tion. A  small  cylindrical  end  is  shown  on  the  bottom  of  the 
screw.  This  represents  the  end  of  the  cylinder  on  which  the 
thread  is  wound. 

Square  Threaded  Screws.  The  square  threaded  screw  is  gen- 
erated by  a  square  revolving  about  the  cylinder  and  at  the  same 


32 

time  moving  in  a  direction  parallel  to  the  axis.  In  the  square 
single  threaded  screw  the  pitch  is  equal  to  the  width  of  a  space 
and  the  thickness  of  a  thread,  measured  in  a  direction  parallel  to 
the  axis. 

Draw  the  projections  of  the  head  and  nut  of  the  same  dimen- 
sions as  in  the  V-threaded  screw.  Lay  off  a  series  of  squares,  the 
sides  of  which  are  equal  to  one-half  the  pitch,  on  the  two  edges  of 
the  screw,  and  find  the  points  in  the  helices  as  in  the  example  pre- 
ceding. It  should  be  observed  that  the  long  curves  show  in  their 
full  lengths,  and  the  short  ones  only  show  to  the  center  in  the 
screws,  whilst  in  the  nut  the  opposite  is  true. 

The  smaller  screws  near  the  center  of  the  plate,  show  how  V 
and  square  threaded  screws  are  after  represented  when  so  small 
that  the  construction  of  the  helices  is  impracticable.  The  con- 
struction only  varies  from  the  larger  ones  inasmuch  as  the  curves 
are  replaced  by  straight  lines. 

Below  there  is  shown  still  another  method  of  representing 
very  small  screws,  either  V  or  square  threaded,  and  the  projection 
of  a  hexagonal  head  with  face  parallel  to  the  plane  of  projection. 

It  is  customary  among  draughtsmen  to  represent  all  threaded 
screws,  unless  very  large,  by  fine  lines  across  the  bolt  representing 
the  points,  and  shorter,  heavier  lines  between,  representing  the 
hollows  of  the  threads. 

Below  is  given  a  table  of  the  Franklin  Institute,  or  United 
States  standard  proportions  for  screw  threads.  This  table  is 
given  here  that  it  may  be  conveniently  referred  to  whenever 
screws  and  nuts  are  to  be  drawn.  A  real  V  thread  is  often  used, 
but  a  thread  very  similar,  having  a  small  flat  part,  in  section,  at 
the  outside  of  each  thread  and  a  similar  flat  part  between  the 
threads,  is  becoming  more  common.  The  dimensions  of  such  a 
thread  are  given  in  the  following  table,  where  diameter  of  screw 
means  the  outer  diameter,  diameter  of  core,  the  diameter  of  the 
cylinder  on  which  the  thread  is  wound,  and  width  of  flat,  the 
width  of  the  flat  part  just  described.  The  four  columns  at  the 
right  relate  to  the  nuts  and  bolt  heads. 


PLATE    VII. 


33 


PROPORTION  OF  SCREW  THREADS,  NUTS  AND  BOLT  HEADS. 


SCRE 

NUTS  AND  HEADS 

U'S. 

HEXAGONAL. 

SQfARE. 

0 

, 

"3                                                                  •£ 

!      ^ 

Diameter 
Sere  \\-  . 

? 

t                  "2                            r-'          5      v.' 

-M            •                                  .          '            *•        .8                               " 

%    £      -t;       2   £      «   t; 

S=        £     «          «     s        s    S 
5O         C    ft           s     c          2      S 

a                        o           £ 

Diagonal. 

Height  t 

4-~  Head. 

1 

i 

20 

.185      .0062               ,»,.. 

5 
1  « 

18 

.240      .0070              },\             jy 

i  i;              «  4 

1 

16 

.294      .0078               sjj              11. 

:!  1 
3  •> 

81 

,Y, 

14 

.344  !    .0089               \»             |i 

-1   1 

L  1  li 

Si 

i 

13 

.400      .0096            1                    • 

M 

r« 

II 

12 

.454       .0104               1,.^                I;' 

1  ft 

ill 

s 

11 

.507      .0113            I,-,           I/,. 

11 

JI 

•I 

10 

.620      .0125            I,',.           1] 

1J 

s 

i 

9 

.731  ;    .0140            Igi           I,-,. 

2A 

ii 

1 

8 

.837      .0156            1J             1J 

2  A 

1  :t 

1  i; 

n 

7 

.940      .0180  i          28:l»           1  }  ;{ 

21 

I* 

VI 

7 

1.065      .0180            2,5,,           2 

9»r 

-S-J 

1 

13 

6 

1.160      .0210            21             2,:!,; 

3  A 

IA 

n 

6 

1.284      .0210            2:l            22 

31 

IA 

M 

51 

1.389      .0227            2};:           L',", 

3| 

1A 

n 

5 

1.490      .0250           3,:!,.,          2:J 

3;!!         12 

ij 

5 

1.615  j    .0250            3J  g           2}^ 

*A       US 

2 

*i 

1.712      .0280            3g              3i 

4lV.               1/6- 

-M 

41     • 

1.9<52       .0280            4,1,.,            31 

4^1          I-- 

21 

4 

2.175      .0310            41             3- 

r>i        i>? 

1 
4 

2.425      .0310            4||           4J 

6             21 

3 

31 

2.628      .0357            5j{             4^ 

fi  !l           2s 

°  i  c;            *^  ,-, 

34 

DRAWING  FROM  ROUGH  SKETCHES. 

Plate  VII  is  given  to  illustrate  the  method  of  making  rough 
sketches  of  an  object  from  which  a  finished  drawing  is  to  be  made. 
The  rough  sketches  here  shown  are  of  a  large  valve  such  as  is  used 
on  large  water  pipes.*  This  example  has  been  chosen  because  it  is 
symmetrical  with  reference  to  the  center  line.  In  such  a  case  as 
this,  it  is  obviously  unnecessary  to  make  complete  sketches  of  the 
whole  object.  Enough  of  the  plan  of  the  object  is  given  above  to 
make  a  complete  plan  from,  in  the  drawing.  The  sketch  of  the 
elevation  below  shows  all  that  is  necessary  for  that. 

In  making  a  rough  sketch,  decide  what  projections  will  best 
represent  the  object,  and  get  in  such  a  position  as  to  see  the  object 
as  nearly  as  possible  as  it  will  appear  in  the  projection,  changing 
the  position  of  observation  for  the  sketches  of  the  different  pro- 
jections. It  must  be  borne  in  mind  that  the  view  a  person  has  of 
an  object  while  sketching  is  a  perspective  view  and  allowance 
must  be  made  for  the  way  it  will  appear  in  projection.  Sketches 
similar  to  the  projections  are  better  than  perspective  sketches  to 
work  from.  The  sketches  should  be  made  in  the  same  relative 
position  that  they  will  appear  in  the  projection  drawing.  Be  sure 
to  represent  every  line  of  the  object  in  the  sketches,  excepting  the 
cases  where  symmetrical  parts  maybe  drawn  from  a  sketch  of  one 
part,  atid  indicate  all  the  dimensions  by  plain  figures  and  arrow 
heads,  taking  all  the  dimensions  possible  from  some  well  defined 
line  like  a  center  line  or  a  bottom  line. 

If  any  of  the  details  on  the  principal  sketch  are  too  small  to 
contain  the  figures  of  the  dimensions,  make  enlarged  sketches  aside 
from  the  other  as  indicated  in  the  plate.  All  that  is  necessary  to 
be  known  about  a  nut  is  the  diameter  of  the  bolt;  the  nut  may 
then  be  constructed  according  to  the  standard. 

Often  a  few  words  of  description  written  on  the  sketch  as,  in 
the  case  of  a  bolt,  four  threads  to  the  inch,  will  describe  a  part 
sufficiently  to  one  acquainted  with  the  standard  proportions  of 
such  common  pieces  as  screw  bolts,  etc. 

One  unaccustomed  to  making  sketches,  is  apt  to  omit  some  di- 
mensions, and  too  great  care  cannot  be  taken  to  have  everv  part 
of  the  object  clearly  indicated  in  some  way  on  the  sketch.  The 
student  should  make  it  a  point  to  endeavor  to  get  all  necessarv 
measurements  on  his  sketch  before  beginning  work  on  his  draw- 
ing, as  in  practice  one  is  often  sent  many  miles  to  obtain  measure- 
ments to  be  worked  up  at  home,  and  an  omission  may  prove  very 
expensive. 


*A  complete  drawing  of  such  an  object  should  show  the  internal 
parts,  but  as  the  object  of  this  plate  is  simply  to  illustrate  the  method  of 
sketching,  the  internal  arrangement  is  not  shown. 


35 


TINTING  AND  SHADING. 


At  this  point  of  the  work  the  following  additional  materials 
will  be  needed : 

A  set  of  water  colors,  a  nest  of  cabinet  saucers,  a  camels  hair 
brush,  a  bottle  of  mucilage  and  brush,  and  a  small  glass  for  water. 

WATER  COLORS. — Winsor  and  Newton's  water  colors  in  "  half 
pans"  are  recommended.  They  should  contain  the  following  colors 
— Burnt  Sienna,  Raw  Sienna,  Crimson  Lake,  Gamboge,  Burnt 
Umber,  Indian  Red  and  Prussian  Blue.  If  the  bottled  ink  has  been 
used  for  the  previous  work,  a  stick  of  India  ink  will  also  need  to 
be  purchased.  All  the  conventional  colors  used  to  represent  the 
different  materials  may  be  mixed  from  the  simple  ones  given  in 
this  list. 

SAUCERS. — The  "nest"  should  contain  six  medium  sized  sau- 
cers. 

BRUSH. — The  camel  hair  brush  should  be  double  ended,  and 
should  be  of  medium  size. 

MUCILAGE. — The  mucilage  needs  to  be  very  thick,  as  it  is  used 
in  shrinking  down  the  heavy  drawing  paper.  The  ordinary  muci- 
lage in  bottles  is  not  fit  for  this  use,  and  it  is  recommended  that 
each  person  buy  the  Gum  Arabic,  and  dissolve  it  in  a  bottle  of 
water,  using  it  as  thick  as  it  will  run. 

WATER  GLASS. — This  glass  is  for  holding  clean  water  with 
which  the  colors  are  mixed.  Any  small  vessel  will  answer  this 
purpose,  but  a  small  sized  tumbler  is  the  most  convenient. 

DIRECTIONS  FOR  SHRINKING  DOWN  PAPER. 

Whenever  a  drawing  is  to  be  tinted  it  must  be  shrunk  down 
in  order  that  it  may  not  wrinkle  after  tinting.  To  shrink  down 
the  paper  proceed  as  follows :  Lay  the  edge  of  the  T-square  paral- 
lel to  an  edge  of  the  paper,  and  about  five-eighths  of  an  inch  from 
it :  and  turn  up  the  paper  at  right  angles,  making  a  sharp  edge 
where  the  paper  is  bent  up  by  pressing  it  hard  against  the  edge  of 
the  T-square  with  the  thumb  nail  or  knife  blade.  Turn  up  all  the 
edges  in  this  wa}"  so  that  the  paper  will  resemble  a  shallow  paper 
box.  The  corners  need  not  be  cut,  through  many  draughtsmen  cut 


36 

out  a  V  shaped  notch  from  each  corner  to  save  trouble  in  folding, 
but  must  be  doubled  over  so  that  all  the  edges  of  the  paper  will 
stand  nearly  perpendicular.  After  this  is  done  the  paper  should  l>e 
turned  over  so  as  to  rest  on  the  upturned  edges,  and  dampened 
ver3*  slightly  with  a  sponge  on  the  back.  Ever\r  part  of  the  paper 
must  be  dampened  except  the  upturned  edges  which  must  lie  kept 
dry  in  order  that  the  mucilage  ma}-  stick.  No  water  should  be 
left  standing  on  the  sheet  when  it  is  turned  over. 

The  paper  should  next  be  turned  over  and  placed  so  that  two 
edges  at  right  angles  may  correspond,  when  turned  down,  to  two 
edges  of  the  drawing  board.  The  other  side  should  then  be  thor- 
oughly wet.  The  mucilage  should  next  be  applied  to  the  dr- 
edges as  rapidly  as  possible.  The  two  edges  that  correspond  to 
the  edges  of  the  drawing  board  should  first  be  turned  down,  great 
care  being  taken  to  leave  no  wrinkles  in  these  edges  nor  in  the  cor- 
ner between  them.  The  other  edges  should  then  be  turned  down. 
the  same  care  being  taken  to  leave  no  wrinkles  either  in  the  edges 
or  corners.  The  edges  must  be  kept  straight,  and,  if  there  are  no 
wrinkles  left  in  the  edges,  the  paper  will  come  down  smooth  when 
dry,  no  matter  how  much  wrinkled  while  wet.  The  natural  shrink- 
age of  the  paper  is  sufficient  without  stretching.  Theedges  should 
be  pressed  down  smooth  with  the  back  of  a  knife  or  the  thumb 
nail,  and  the  paper  should  be  allowed  to  drj*  slowly.  Paper  should 
never  be  dried  in  the  sun  or  in  artificial  heat  as  it  will  get  too  dry 
and  afterwards  become  loose  when  exposed  to  ordinao*  temper- 
ature. 

Considerable  practice  may  be  necessary  before  the  pa]>er  can 
be  shrunk  down  successfully,  but  if  the  directions  above  are  follow- 
ed closely  there  need  be  no  difficulty.  The  paper  must  be  dampen- 
ed evenly,  and  the  mucilage  must  lie  put  on  evenly  and  abundant- 
ly. Great  care  must  be  taken  not  to  drop  any  mucilage  on  the 
middle  of  the  drawing  board,  and  not  to  get  any  beyond  the  dry 
edge  of  the  paper.  Otherwise  the  paper  may  be  stuck  down  so  as 
to  make  trouble  in  cutting  the  plate  off  when  finished. 

THE  PLATING   IN   TINTING  AND   SHADING. 

Plates  A  and  B,  which  should  l>e  prepared  by  the  Instructor 
and  hung  on  the  walls  of  the  Drawing  Room,  contain  the  most 
common  forms  that  are  brought  out  by  shading  in  ordinary  me- 
chanical drawings  and  the  most  common  conventional  colors  used 
in  working  drawings.* 


"These  colored  plates  could  not  conveniently  be  placed  in  this  pamphlet, 
and  in  cases  where  access  cannot  be  had  to  the  wall  plates  here 
mentioned,  it  is  recommended  that  the  instructor  make  similar  ones  for 
the  use  of  the  students.  Plate  A  contains  five  rectangles  in  the  upper 
ro\\,  the  first  three  of  which  are  to  be  plain  shades,  and  the  other  two 


37 

Plate  A,  which  is  shaded  altogether  with  India  ink  is  to  be 
done  first. 

Shrink  down  a  half  sheet  of  Imperial  paper,  and  mark  it  in- 
side of  the  edges  so  that  it  may  be,  when  cut  off,  twenty  by 
thirteen  and  a  half  inches.  Lay  out  a  border  one  inch  inside  of 
the  lines  just  drawn  and  draw  the  outlines  of  the  figures  with 
a  very  sharp  pen,  using  the  best  of  ink,  and  njaking  the  lines  as 
fine  as  possible.  The  figures  must  be  drawn  of  the  same  size, 
and  arranged  the  same  way  as  in  the  wall  plates.  The  border 
and  the  letters,  which  are  to  correspond  with  those  in  the  wall 
plates,  should  not  be  drawn  until  the  plate  is  shaded.  The 
dimensions  of  the  figures  need  not  be  put  on  to  the  finish  plates 
at  all. 

After  inking  in  the  figures,  the  plate  should  be  washed  over 
with  clean  water  to  take  out  any  surplus  ink  and  to  leave  the 
paper  in  better  condition  for  the  water  shades.  The  paper  should 
be  sopped  very  lightly  with  a  sponge  and  a  large  quantity  of 
water  should  be  used.  After  washing  the  paper  allow  it  to  dry 
slowly.  If  the  paper  is  dried  in  the  sun  it  will  get  so  warm  that 
the  shades  will  dry  too  rapidly.  When  the  paper  is  down  smooth 
and  dry,  it  should  be  placed  on  the  drawing  table  slightly  in- 
clined in  one  direction  in  order  that  the  ink  or  water  color  may 
always  flow  in  one  direction. 

Take  a  saucer  half  full  of  clean  water  and  by  rubbing  the 
wet  brush  on  the  end  of  a  stick  of  ink  mix  enough  India  ink  to 
make  a  shade  no  darker  than  that  in  (c),  on  wall  plate.  A  small 
piece  of  paper  should  be  kept  to  try  the  shades  on  before  putting 
them  on  the  plate.  Mix  the  ink  thoroughly  with  one  end  of  the 
brush  before  applying  to  the  paper.  One  end  of  the  brush  should 
always  be  used  in  the  ink  or  tint  while  the  other  end  is  kept 
clean  for  blending. 

With  considerable  ink  in  the  brush  but  not  nearlj'  all  it  will 
hold,  commence  at  the  top  line  of  (a),  and  follow  it  carefully  with 
the  first  stroke.  Before  the  ink  dries  at  the  top,  lay  on  the  ink 
below  by  moving  the  brush  back  and  forth,  using  enough  ink  in 
the  brush  so  that  it  will  flow  gradually,  with  the  help  of  the 
brush,  toward  the  bottom.  The  lines  must  be  followed  carefully 
at  first,  and  the  brush  should  not  be  used  twice  over  the  same 
place.  In  following  a  line  w5th  the  brush  get  in  such  a  position 
that  the  forearm  will  be  perpendicu'ar  to  the  direction  of  the  line. 


are  to  be  blended.  The  lower  row  of  figures  in  this  plate  contains 
plans  and  elevations  of  the  following  figures  in  the  order  named,  a 
prism,  pyramid,  cylinder,  cone  and  sphere.  Plate  B  contains,  in  the 
upper  row,  circular  figures  tinted  to  represent  the  conventions  for 
cast-iron,  wrought-iron,  steel  and  brass  ,  and  in  lower  row  four  square 
figures  with  the  conventional  colors  for  copper,  bricl<,  stone  and  wood. 


38 

Do  not  paint  the  shades  on  but  allow  them  to  flow  quite  freely 
after  the  brush.  In  shading  or  tinting  there  is  great  danger  of 
making  clouded  places  and  "water  lines"  unless  the  greatest  of 
care  is  taken  in  using  the  brush.  If  the  brush  is  used  over  a  shade 
that  is  parti}-  dry  it  will  make  it  clouded.  And  if  the  edge  of  the 
shade  is  allowed  to  dry  before  finishing,  a  "water  line"  is 
produced  where  the  new  shade  is  joined  to  the  old. 

In  finishing  up  a  figure  the  ink  should  be  taken  up  with  the 
brush  so  that  it  will  not  spread  beyond  the  lines.  The  sun  should 
never  be  allowed  to  shine  on  the  paper,  as  it  will  dry  it  too  fast. 
A  damp  day  is  better  for  tinting  or  shading  than  a  dry  one  for 
the  reason  that  the  drying  is  then  very  slow.  The  shades  of  (a ), 
(b)  and  (c)  are  all  plain.  Commence  on  (a),  and  while  it  is  dry- 
ing put  a  coat  on  (b).  To  determine  when  a  shade  is  dry,  look  at 
it  very  obliquely,  and  if  it  does  not  glisten  it  is  ready  for  another 
coat.  Put  four  coats  on(  a),  two  on  (b)  and  one  on  (c). 

BLENDING. — A  varying  shade,  such  as  is  noticed  in  viewing  a 
cylindrical  object,  may  be  obtained  by  blending  with  India  ink. 
This  operation  of  blending  is  emploj'ed  in  bringing  out  the  forms 
of  objects,  as  seen  in  the  lower  figures  of  plate  A. 

The  figures  (d)  and  (e)  are  for  practice  in  blending  before  ap- 
plying to  the  solid  object  below.  Begin  (d)  by  laying  on  a  flat 
shade  about  an  eight  of  an  inch  wide,  using  but  little  ink ;  and 
when  nearly  dry  take  the  other  end  of  the  brush,  slightly  moisten- 
ed in  clean  water,  and  run  it  along  the  lower  edge  of  the  shade 
blending  downward.  When  this  is  entirely  dry,  lay  on  another 
plain  shade  a  little  wider  than  the  first,  and  blend  it  downward 
in  the  same  way.  Use  but  little  water  and  lay  on  the  shade  in 
strips,  always  commencing  at  the  top  line.  When  finished  the 
lower  part  will  have  had  but  one  coat  whilst  the  upper  part  will 
have  had  several.  Blend  (c)  in  the  same  way  as  (d),  but  use  nar- 
rower strips  of  tint  in  order  to  make  more  contrast  between  the 
top  and  bottom. 

SHADING  SOLIDS.— When  a  solid  object  is  placed  in  a  strong 
light  coming  principally  from  one  direction,  a  strong  contrast 
will  be  noticed  between  the  shades  of  the  different  portions,  and 
these  shades  serve  to  reveal  the  shape  of  the  object  much  more 
clearly  than  when  it  is  placed  in  diffused  light  only.  For  this 
reason,  as  well  as  from  the  fact  that  the  laws  of  the  shades  of  an 
object  in  light  from  one  direction  are  very  simple,  the  shades  in 
a  drawing  are  usually  made  to  correspond  to  those  of  a  body 
where  the  light  comes  from  a  single  window.  In  all  cases, 
however,  it  is  assumed  that  there  is  a  certain  amount  of  diffused 
light,  such  as  is  always  present  in  a  room  lighted  by  a  single 


39 

window,  aside  from  the  strong  beam  of  light  that  conies  directly 
through  the  window. 

1.  The  shades  of  an  object  arc  always  in  greater  contrast 
when  the  object  is  near  the  eye  than  far  away. 

2.  The  lightest  portion  of  a  cylinder,  cone  or  sphere  is  where 
the  direct  light  strikes  the  object  perpendicularly,  and  the  darkest 
portion  of  the  same  is   where   the  light   strikes  tangent   to   the 
object,  the  shade  varying  gradually  between  these  parts. 

The  facts  just  given  may  easily  be  proved  by  holding  a  body 
in  the  light  and  noticing  the  shades. 

These  facts  we  will  assume  as  the  principles  that  govern  the 
shading  of  the  following  objects. 

In  view  of  the  above  principles  the  first  thing  to  be  deter- 
mined, after  assuming  the  direction  of  the  light,  is  where  the 
lightest  and  darkest  parts  will  be,  and  what  parts  are  near  to 
the  observer  and  what  parts  are  farthest  away.  In  all  the  fol- 
lowing cases  we  shall  assume  the  light  to  come  from  over  the  left, 
shoulder,  make  the  angle  of  forty-five  degrees  with  both  the 
vertical  and  horizontal  planes  of  projection. 

THE  PRISM. — By  the  use  of  the  forty-five  degree  triangle  on 
the  T  square,  draw  the  arrows  as  shown  on  the  plan.  The 
points  where  these  touch  the  plan  show  where  the  direct  light 
will  strike  by  the  prism.  By  dropping  verticals  from  these  points 
we  see  that  one  will  fall  behind  the  elevation  and  one  in  front, 
showing  that  the  vertical  edge  near  the  right  separates  the  light 
from  the  dark  portions  of  the  prism.  The  light  will  nowhere 
strike  the  prism  perpendicularly,  but  it  will  strike  that  face  near- 
est the  left  the  most  directly  of  any,  and  it  will,  of  course,  be  the 
lightest  face  of  the  prism.  The  front  face  will  be  a  little  darker, 
and  the  right-hand  face,  being  lighted  only  by  diffused  light,  will 
lie  much  darker  than  either  of  the  other  two. 

The  plan  showing  only  the  upper  base,  receives  light  at  the 
same  angle  as  the  front  face,  and  will  have  the  same  shade,  which 
should  be  about  the  same  as  on  the  plate,  and  not  blended. 
Considering  the  principle  that  the  contrast  is  less  between  light 
and  shade  at  a  distance,  we  know  that  the  outside  parts  of  the 
faces  on  the  right  and  left  will  tend  to  assume  nearly  the  same 
shade  as  they  recede  from  the  observer,  consequently  the  light 
face  should  be  blended  slight!}-  toward  the  right,  and  the  dark 
face  on  the  right  should  also  be  blended  toward  the  right,  making 
the  former  darker  toward  the  outside  and  the  latter  lighter 
toward  the  outside. 

THE  PYRAMID. — The  light  and  dark  portions  are  found  in  the 
same  way,  and  the  elevation  is  shaded  almost  precisely  like  the 


40 

prism.  The  top  recedes  slightly  and  the  contrast  there  should  be 
slighth-  less  than  at  the  bottom  where  it  is  nearer  the  eye.  The 
faces  in  the  plan  recede  very  rapidly,  and  the  greatest  contrast 
must  be  at  the  top.  The  upper  right  hand  part  receives  only  dif- 
fused light. 

THK  CYLINDER. — The  dotted  lines  show  011  the  plate  the 
method  of  finding  the  lightest  and  darkest  portions.  Use  the 
forty -five  degree  triangle  on  the  T  square  so  as  to  draw  the  two 
diagonal  radii  as  shown.  Where  the  one  on  the  right  cuts  the 
lower  semi-circumferences  is  the  darkest  point  and  where  the 
other  cuts  the  same  on  the  left  is  the  lightest  point.  These  points 
projected  down  will  give  the  lightest  and  darkest  lines  on  the 
elevation.  Blend  quite  rapidly  both  ways  from  the  dark  line  and 
toward  the  right  from  the  left  hand  limiting  elements.  The 
shades  near  the  limiting  elements  should  be  about  alike  on  the 
two  sides. 

The  shade  of  the  plan  should  be  the  same  as  that  of  the  plan  of 
the  prism. 

THE  CONE. — The  instructions  given  for  shading  the  cylinder 
with  those  given  for  shading  the  pyramid  apply  to  this  figure. 
Great  care  must* be  taken  to  bring  out  the  vertex  in  the  plan. 

THE  SPHERE.— It  will  be  evident  on  consideration  that  the 
darkest  portion  of  the  sphere  is  a  great  circle,  the  plane  of  which 
is  perpendicular  to  the  direction  of  the  light ;  but,  as  this  great 
circle  is  not  parallel  to  the  planes  of  projection,  its  projections  are 
both  ellipses.  There  can  evidently  be  but  one  point  where  the 
light  can  strike  the  sphere  perpendicularly,  and  that  is  where  the 
radius  parallel  to  the  direction  of  the  light  meets  the  surface. 

To  find  the  lightest  points  in  plan  and  elevation,  join  the  cen- 
ters of  the  plan  and  elevation  by  a  vertical,  draw  a  diameter  of 
the  plan  upward  and  to  the  right  at  an  angle  of  fortA'-five  degrees 
and  of  the  elevation  downward  and  to  the  right.  Draw  a  line 
using  the  forty-five  degree  triangle  from  the  point  where  the  line 
joining  the  centers,  cuts  the  circumference  of  the  plan*  perpendicu- 
lar to  the  diameter  drawn  in  the  same.  Where  it  intersects  the 
same  will  be  the  lightest  point  in  the  plan;  a  line  similarly  drawn 
gives  the  lightest  point  in  the  elevation  as  shown  in  the  plate. 
The  ellipse,  which  is  the  dark  line  of  the  object,  crosses  the  two  di- 
ameters drawn  on  the  plan  and  elevation,  just  as  far  from  the  cen- 
ters of  each  as  the  light  points  are  from  the  same.  These  points 
may  be  laid  off  from  the  centers  by  means  of  the  dividers.  The 
shade  of  the  plan  and  elevation  of  the  sphere  are  exactly  alike,  but 
the  position  of  the  light  and  dark  portions  are  different,  as  seen 
in  the  plate. 


41 

Commence  by  laying  on  a  narrow  strip  of  shade  in  the  form 
of  an  half  ellipse  over  the  dark  point.  Add  other  and  wider 
strips  of  the  same  general  form,  and  blend  each  toward  the  light 
point  toward  the  outside.  Great  pains  must  be  taken  with  this 
to  get  the  correct  shades  and  the  two  exactly  alike. 

Put  on  all  the  lines  and  letters  shown  on  the  plate,  making 
the  dotted  lines  and  arrows  very  fine. 

TINTING. — Plate  B  contains  the  conventional  tints  for  the  fol- 
lowing materials :  Cast  Iron,  Wrought  Iron,  Steel,  Brass,  Copper, 
Brick,  Stone  and  Wood.  The  square  figures  are  three  inches  on  a 
side,  and  the  circular  figures  have  diameters  of  the  same  length. 
These  colors  are  more  difficult  to  lay  on  evenly  than  the  India  ink 
shades,  but  what  has  been  said  about  the  application  of  ink 
shades  applies  to  them.  Great  pains  must  be  taken  to  have  the 
paperingood  condition,  and  to  keep  the  colors  well  mixed.  Enough 
color  should  be  mixed  to  finish  the  figure  as  it  is  almost  impossi- 
ble to  match  the  colors  exactl3r.  Wash  the  brush  thoroughly  be- 
fore commencing  a  new  figure.  The  plates  are  to  be  lettered  like 
the  wall  plate,  the  initials  at  the  bottom  standing  for  the  colors 
used.  Below  are  given  the  materials  to  be  used  in  each  convention, 
the  exact  proportions  of  these  can  best  be  found  by  experiment, 
comparing  the  colors  with  those  on  the  wall  plate.  A  number  of 
thin  coats,  well  laid  on,  generally  look  more  even  than  when  the 
tints  are  laid  on  in  single  coats. 

For  Cast  Iron,  use  India  ink,  Prussian  Blue  and  Crimson 
Lake;  for  Wrought  Iron,  Prussian  Blue  and  India  ink;  for  Steel, 
Prussian  Blue ;  for  Brass,  Gamboge,  Burnt  Umber  and  Crim- 
son Lake ;  for  Copper,  Crimson  Lake  and  Burnt  Umber ;  for  Brick, 
Indian  Red;  for  Stone,  India  ink  and  Prussian  Blue;  for  Wood, 
Raw  and  Burnt  Sienna.  The  convention  for  the  body  of  wood  is 
made  by  laying  on  a  light  coat  of  Raw  Sienna,  and  the  Grain  is 
made  by  applying  the  Burnt  Sienna,  after  the  first  is  dry,  with  the 
point  of  the  brush,  blending  slightly  in  one  direction.  Other  com- 
binations requiring:  different  colors  are  often  used  but  as  they  are 
all  conventional  the  above  will  serve  as  an  illustration. 


42 


SHADE  LINING. 

Since  the  Blue  Print  method  of  copying  drawings  has  come  in- 
to use,  very  little  tinting  and  shading  with  the  brush  is  done,  but 
instead  the  form  is  often  indicated  by  line  shading.  In  many 
cases  therefore  the  plates  of  tinting  and  shading  may  be  omitted, 
but  in  that  case  the  student  should  study  carefully  the  directions 
on  pages  39  and  40,  as  to  the  method  of  finding  light  and  dark 
places  and  apply  the  principles  to  shade  lining. 

Shade  lining  is  a  method  of  representing  the  shades  of  an  ob- 
ject by  a  series  of  lines  drawn  on  the  projections  so  as  to  produce 
the  same  general  effect  as  when  blended  with  Indiaink.  This  effect  is 
produced  by  making  the  lines  very  fine  and  at  a  considerable  dis- 
tance apart  on  the  light  portions,  and  quite  heav3r  and  close  to- 
gether on  the  dark  portions.  This  method  of  shading  is  often 
employed  in  uncolored  drawings  to  bring  out  the  forms  of  parts 
that  might  not  otherwise  be  clearly  understood,  and  often  to  give 
a  drawing  a  fine  finished  appearance. 

Plates  C  and  D  contain  the  figures  that  generally  require  to  be 
shaded  in  ordinary  drawings.  Each  of  these  plates  is  made  on  a 
quarter  sheet  of  Imperial  paper,  and  of  the  size  shown. 

These  plates  are  made  smaller  than  the  others  for  the  reason 
that  each  figure  contains  a  great  many  lines. 

It  is  recommended  that  a  half  sheet  of  Imperial  paper  be 
shrunk  down, and  that  this  be  divided  when  the  plates  are  finished. 

The  dimensions  of  each  figure  are  marked  on  the  plates  but 
need  not  be  marked  on  those  drawn  by  students. 

The  figures  should  all  be  penciled  in  of  the  dimensions  indi- 
cated, and  in  the  positions  shown  in  the  plates.  The  limiting  lines 
of  the  figures  to  be  shade  lined  should  be  drawn  as  fine  as  possible, 
and,  on  the  practice  figures,  had  better  not  be  inked  until  the 
shade  lines  are  drawn. 

The  light  and  dark  portions  are  found  as  in  Plate  A,  prev- 
iously referred  to,  and  as  the  shades  to  be  represented  are  the  same 
as  in  that  plate,  reference  is  made  to  the  remarks  on  shades  in 
the  description  of  the  same. 

Xo  description  can  lie  of  so  much  value  as  a  thorough  study  of 


43 

the  plates.  Notice  carefully  the  gradation  of  the  shade  lines  on 
each  projection.  In  Plate  C,  the  shade  lines  are  all  parallel;  but  in 
Plate  D,  they  are  neither  parallel  nor  of  the  same  width  through- 
out. 

In  the  drawing  of  the  sphere  the  light  lines  are  made  full  circles, 
with  the  lightest  point  in  each  projection  as  a  center.  The  middle 
portions  only  of  dark  lines  are  made  with  the  compasses,  the  ends 
being  finished  free  hand.  In  shade  lining  the  pen  must  be  kept  very 
sharp  and  the  ink  must  run  well.  Put  on  all  the  arrows  and 
dotted  lines  shown  on  the  plates. 

In  practice,  except  in  very  fine  drawings,  it  is  customary  to 
use  much  fewer  lines  than  are  here  shown,  putting  on  just  enough 
to  indicate  the  form  rather  than  to  fully  show  it.  As,  however, 
one  able  to  do  the  work  here  shown  can  easily  modify  the  method 
for  himself,  it  is  thought  best  to  show  how  to  do  the  best  work. 


rS 


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